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Get Free NCERT Solutions for Class 12 Mathematics Chapter 7 Integrals, set up after careful examination by profoundly experienced Mathematics instructors, at Swastik Classes. NCERT Solutions are extremely helpful while doing your homework and also for your Class 12 board exam preparation. We have given bit-by-bit answers to every one of the questions given in the NCERT class Mathematics course reading. This solution is free to download and the questions are systematically arranged for your ease of preparation and in solving different types of questions. To score good marks, students are encouraged to get familiar with these NCERT solutions of Chapter 7 Integrals.

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Answers of Mathematics NCERT solutions for class 12 Chapter 7 Integrals

Chapter 7

Integrals

Exercise-7.1

Question 1:

sin 2x

Answer:

The anti-derivative of sin 2x is a function of x whose derivative is sin 2x.

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7381/NCERT_Solution_Math_Chapter_7_final_html_m697e7f3e.gif

Therefore, the anti-derivative of is

Question 2:

Cos 3x

Answer:

The anti-derivative of cos 3x is a function of x whose derivative is cos 3x.

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7383/NCERT_Solution_Math_Chapter_7_final_html_37502309.gif

Therefore, the anti-derivative of  is .

Question 3:

e2x

Answer:

The anti-derivative of e2is the function of x whose derivative is e2x.

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7385/NCERT_Solution_Math_Chapter_7_final_html_m28a7365e.gif

Therefore, the anti-derivative of is  

Question 4:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7386/NCERT_Solution_Math_Chapter_7_final_html_d228bea.gif

Answer:

The anti-derivative of

is the function of whose derivative is 

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7386/NCERT_Solution_Math_Chapter_7_final_html_m22680595.gif

Therefore, the anti-derivative of is  

Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7387/NCERT_Solution_Math_Chapter_7_final_html_m64ac22d.gif

Answer:

The anti-derivative of  is the function of x whose derivative is .

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7387/NCERT_Solution_Math_Chapter_7_final_html_37d39277.gif

Therefore, the anti-derivative of  is .

Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7388/NCERT_Solution_Math_Chapter_7_final_html_m1fde19c0.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7388/NCERT_Solution_Math_Chapter_7_final_html_m1fde19c0.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7388/NCERT_Solution_Math_Chapter_7_final_html_55f021fe.gif

Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7389/NCERT_Solution_Math_Chapter_7_final_html_76cc6ec4.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7389/NCERT_Solution_Math_Chapter_7_final_html_76cc6ec4.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7389/NCERT_Solution_Math_Chapter_7_final_html_5e8acc03.gif

Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7390/NCERT_Solution_Math_Chapter_7_final_html_5ebb449b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7390/NCERT_Solution_Math_Chapter_7_final_html_5ebb449b.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7390/NCERT_Solution_Math_Chapter_7_final_html_146e5d64.gif

Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7391/NCERT_Solution_Math_Chapter_7_final_html_4b2d046b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7391/NCERT_Solution_Math_Chapter_7_final_html_4b2d046b.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7391/NCERT_Solution_Math_Chapter_7_final_html_1319c4b3.gif

Question 10:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7392/NCERT_Solution_Math_Chapter_7_final_html_m21b097a0.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7392/NCERT_Solution_Math_Chapter_7_final_html_m21b097a0.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7392/NCERT_Solution_Math_Chapter_7_final_html_353d003.gif

Question 11:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7393/NCERT_Solution_Math_Chapter_7_final_html_m45687b1f.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7393/NCERT_Solution_Math_Chapter_7_final_html_m45687b1f.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7393/NCERT_Solution_Math_Chapter_7_final_html_m60c06e96.gif

Question 12:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7394/NCERT_Solution_Math_Chapter_7_final_html_m352a7734.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7394/NCERT_Solution_Math_Chapter_7_final_html_m352a7734.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7394/NCERT_Solution_Math_Chapter_7_final_html_75ad4a9d.gif

Question 13:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7395/NCERT_Solution_Math_Chapter_7_final_html_m762c7093.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7395/NCERT_Solution_Math_Chapter_7_final_html_m762c7093.gif

On dividing, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7395/NCERT_Solution_Math_Chapter_7_final_html_19f077a8.gif

Question 14:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7396/NCERT_Solution_Math_Chapter_7_final_html_1b1677ba.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7396/NCERT_Solution_Math_Chapter_7_final_html_1b1677ba.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7396/NCERT_Solution_Math_Chapter_7_final_html_982111b.gif

Question 15:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7397/NCERT_Solution_Math_Chapter_7_final_html_ef118c8.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7397/NCERT_Solution_Math_Chapter_7_final_html_ef118c8.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7397/NCERT_Solution_Math_Chapter_7_final_html_2b7fadb0.gif

Question 16:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7398/NCERT_Solution_Math_Chapter_7_final_html_70eedab9.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7398/NCERT_Solution_Math_Chapter_7_final_html_70eedab9.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7398/NCERT_Solution_Math_Chapter_7_final_html_m155aa073.gif

Question 17:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7399/NCERT_Solution_Math_Chapter_7_final_html_m3513f78e.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7399/NCERT_Solution_Math_Chapter_7_final_html_m3513f78e.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7399/NCERT_Solution_Math_Chapter_7_final_html_552b2cfe.gif

Question 18:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7400/NCERT_Solution_Math_Chapter_7_final_html_55f8b69e.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7400/NCERT_Solution_Math_Chapter_7_final_html_55f8b69e.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7400/NCERT_Solution_Math_Chapter_7_final_html_m6170702b.gif

Question 19:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7401/NCERT_Solution_Math_Chapter_7_final_html_69cfa7d9.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7401/NCERT_Solution_Math_Chapter_7_final_html_m5d1cbb50.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7401/NCERT_Solution_Math_Chapter_7_final_html_m8779d63.gif

Question 20:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7402/NCERT_Solution_Math_Chapter_7_final_html_m4223465.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7402/NCERT_Solution_Math_Chapter_7_final_html_m4223465.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7402/NCERT_Solution_Math_Chapter_7_final_html_m63d2802c.gif

Question 21:

The anti-derivative of equals

(A) (B) 

(C)  (D) 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7403/NCERT_Solution_Math_Chapter_7_final_html_m53d4ecad.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7403/NCERT_Solution_Math_Chapter_7_final_html_7839ddd7.gif

Hence, the correct answer is C.

Question 22:

If such that f(2) = 0, then f(x) is

(A)   (B) 

(C)   (D) 

Answer:

It is given that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7404/NCERT_Solution_Math_Chapter_7_final_html_3ad53d45.gif

∴Anti derivative of 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7404/NCERT_Solution_Math_Chapter_7_final_html_m38dfbbfb.gif

Also,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7404/NCERT_Solution_Math_Chapter_7_final_html_m601912de.gif

Hence, the correct answer is A.

Exercise-7.2

Question 1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_2daa2dc2.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_44a7d5a8.gift

∴2x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_285bfdcb.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_4f3f798c.gif

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7406/NCERT_Solution_Math_Chapter_7_final_html_4fda87ef.gif

Answer:

Let log |x| = t

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7406/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7406/NCERT_Solution_Math_Chapter_7_final_html_52ef2a81.gif

Question 3:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_m61551e94.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_m4370fbad.gif

Let 1 + log t

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_m2ebe9270.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_1cd43b92.gif

Question 4:

sin x ⋅ sin (cos x)

Answer:

sin x ⋅ sin (cos x)

Let cos x = t

∴ −sin x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7408/NCERT_Solution_Math_Chapter_7_final_html_69fb365b.gif

Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_42180c1b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_1400fe22.gif

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_m6b8fb0e5.gif

∴ 2adx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_396f0375.gif

Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m6a9188e.gif

Answer:

Let ax + b = t

⇒ adx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m5727662a.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m10c63328.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m31dc4753.gif

Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7414/NCERT_Solution_Math_Chapter_7_final_html_6cb9b43.gif

Answer:

Let 

∴ dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7414/NCERT_Solution_Math_Chapter_7_final_html_4c24a780.gif

Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7416/NCERT_Solution_Math_Chapter_7_final_html_4e827515.gif

Answer:

Let 1 + 2x2 = t

∴ 4xdx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7416/NCERT_Solution_Math_Chapter_7_final_html_m34de4fbb.gif

Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_m4b12630c.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_m7c0ef6b.gif

∴ (2x + 1)dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_76f6bdb0.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_m75db78a0.gif

Question 10:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_520c9935.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_4563a36f.gif

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_m3aa49064.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_mf4b4123.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_2d5d2240.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_1a4ba108.gif

Question 11:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7422/NCERT_Solution_Math_Chapter_7_final_html_44d0f94f.gif

Answer:

UhQrUw0DR7F7q3fKBQwNAq1pqlCEOFgclEsOF7FbJy9s NIEQklOClt8j ejU2Lca61QqMIh9edKLjn7SddN8HMQpuHichGBnF2nI9Lptn1ev7VoLO68JXBm4 TusnRNKBpsk5s

Put

rFTl5enKrsr7bBa 4mLGWKpqdsfQc78EVgIBstqrRsyTWuIzK1AHoSzVJBreqZwgDY9iibpIbPC2AM2u7Fkf5VY55Y2ABvbfYSBd4pS3iHDeqLNBA7CZKcTyT4fR2uj yW0LRdM
VDTHQnvyVsWWyskZGGW12x5hXUQp6ZdnCzK2riaUzx8MoPswjBW FgFOK9O0yojst9NtNcWl KoDJVIKUEQyQyA6Hp9xw3zaYikJHbF99CQ tvL4ZwKCOjTOqh ggJ IMZ5wnA
4PMXVzRQ3OatI9q qC8egydIzbiKr4x2IUYuPcW9p6ZUJzTQ 6lszOWa 2fl8FnXaPFswAlvyW4f4ewDaa3GBPVOTNSrMDPQBTJ zZthQ6kwnOKhdA AfvOAHMnXDuBfZLv Fp4
jc9rWCwTS5Gbg02nIrBUU6YwsPeUz hX52qIeDY4G eVbCk i7 FlK5rgmmeZ6IhwbwLPoSpTuEHQjaAho2JNd2Mz23FT MxoVNwLobF7s7l3UT8yW8ugtsFBh 0NohIQf k8s
Zl263Eno1N18TQl0rfGdLJOZImUoIyqacRo IR3wdLB2KD0aVoLLf Ug41qrwdO1 CiUju9YC8UuYbSTOPkz8kxIH410v HciLjqXDTj7am y pvRMfmaoMGvHNvgAXAXGSbXLM
GJexfn8BlezXbO4 Djf4H2NdL0YWAtJXHs4iDZkXohbzkHqDVbactghLeMU6vMMxbmnSuEJoXyV7q6eD23d8PJm9OtwgyTEFI218r yHTk9Ug G3eaoJDddOSvHzipjb Qn6Sg
rI r34t81znakG1X3G07PIeBnUWRUVUPez949RkrHb3TrUXjnpmP5IDLS91RLA8ZVh7llBtu qwiqkGePISIFPX zNLXB6XLET6mTONTmPvk9n2SSJG

Question 12:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_m132ed99c.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_6e628098.gif

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_27cd530.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_m452c22f6.gif

Question 13:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7427/NCERT_Solution_Math_Chapter_7_final_html_3d95f002.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7427/NCERT_Solution_Math_Chapter_7_final_html_26539378.gif

∴ 9x2 dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7427/NCERT_Solution_Math_Chapter_7_final_html_m649abaa8.gif

Question 14:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_78cb1997.gif

Answer:

Let log x = t

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_3e19e712.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_a99ad2.gif

Question 15:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7442/NCERT_Solution_Math_Chapter_7_final_html_m42ca294.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7442/NCERT_Solution_Math_Chapter_7_final_html_m7516177.gif

∴ −8x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7442/NCERT_Solution_Math_Chapter_7_final_html_m15aecbfc.gif

Question 16:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7444/NCERT_Solution_Math_Chapter_7_final_html_2939b26a.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7444/NCERT_Solution_Math_Chapter_7_final_html_5bcc50a3.gif

∴ 2dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7444/NCERT_Solution_Math_Chapter_7_final_html_m513791a7.gif

Question 17:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_6ddfda2d.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_m2b929949.gif

∴ 2xdx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_m650f6bc9.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_7c9ba91.gif

Question 18:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m481a6bf9.gif

Answer:

Let 

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m26ae7d6c.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_277fdce0.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m666c0c5.gif

Question 19:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_4bdb887f.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_4bdb887f.gif

Dividing numerator and denominator by ex, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_m6cb48a41.gif

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_c22a63c.gif

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_549b934c.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_m449ce1c0.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_79d625cd.gif

Question 20:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m8dd01a5.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m67028847.gif

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m3d96998a.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m3314c039.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m5a357383.gif

Question 21:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7456/NCERT_Solution_Math_Chapter_7_final_html_60990151.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7456/NCERT_Solution_Math_Chapter_7_final_html_60990151.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7456/NCERT_Solution_Math_Chapter_7_final_html_50852ccc.gif
Hi1xii3G5aNMTnotmLYI5FfjISzVLm5IBnGqNGfzyPHPdvr8F AfaR3pDDyieQ4fWWDJXjPOEL50nTFjz8GyrYxZJmw07tLrqtpeR qG3seIFa zJs2jPco fKjV6uu980lE5UQ
UjUiRpx 6MANfjr8aOVMv fVk3YlB plJNS3EkcffQkfKPCXPYubZSd6v50KOdo6gz1EzcOiBpO eE9CxtqO4VWcBUXptHKOE7WeGX k8NZ ikzb w2lkEDZ7b AvJ579ORWuOI

Put

wA05YFTuGaUUHoHU2JeEcZKZ9YYVw7v7WBxYJcAf3xIkxTog ik b3ThCBfuPpd7wr Y3xRAO47xO2AkNB6OQqFwVKr uPjjPlFY6XK5tpFiYH kvSd5UimaEpkn0C6YUAtnF5U
rmqTZtc4Q8apNrR PTQDNgCxbt IxHt30wnkJ13bM4oAAoRGT4GvD5jcm5UhulI2E4qGXNgvwiVj81GM T4wG Uo5vM JlyXDXiJf7uTZpVUCXWdMU1dekgJiXwN4O6NCbdn7S0
qaFxYr6ErHn1tmLKOAVJrRwCmEXs0iCKXkcrOM6lhWjTHtGwnwyo JAvNTp3GJ6HLIc60RfPMy0

Question 22:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7458/NCERT_Solution_Math_Chapter_7_final_html_30c46dae.gif

Answer:

Let 7 − 4x = t

∴ −4dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7458/NCERT_Solution_Math_Chapter_7_final_html_5dbac121.gif

Question 23:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_59d50245.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_m127d1393.gif

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_45979e45.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_d934bc5.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_m615ee39c.gif

Question 24:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_m1e861f.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_m1b300ce0.gif

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_1d11b643.gif

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_6bb65edd.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_29ad5f75.gif

Question 25:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_m67c449bc.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_6fe644c1.gif

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_m589b30d9.gif

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_ad32e6e.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_56990d0f.gif

Question 26:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_m3f7ab82.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_m7e7680c5.gif

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_mf4b4123.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_1195f5d9.gif

Question 27:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7480/NCERT_Solution_Math_Chapter_7_final_html_396850ae.gif

Answer:

Let sin 2x = t

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7480/NCERT_Solution_Math_Chapter_7_final_html_ad1a990.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7480/NCERT_Solution_Math_Chapter_7_final_html_103e3be3.gif

Question 28:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7483/NCERT_Solution_Math_Chapter_7_final_html_m24b03964.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7483/NCERT_Solution_Math_Chapter_7_final_html_589aee2d.gif

∴ cos x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7483/NCERT_Solution_Math_Chapter_7_final_html_4631f4db.gif

Question 29:

cot x log sin x

Answer:

Let log sin x = t

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7485/NCERT_Solution_Math_Chapter_7_final_html_m3e62cda.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7485/NCERT_Solution_Math_Chapter_7_final_html_8ef846c.gif

Question 30:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7487/NCERT_Solution_Math_Chapter_7_final_html_5910cc29.gif

Answer:

Let 1 + cos x = t

∴ −sin x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7487/NCERT_Solution_Math_Chapter_7_final_html_537be00f.gif

Question 31:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7489/NCERT_Solution_Math_Chapter_7_final_html_594e5e18.gif

Answer:

Let 1 + cos x = t

∴ −sin x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7489/NCERT_Solution_Math_Chapter_7_final_html_m24732181.gif

Question 32:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7490/NCERT_Solution_Math_Chapter_7_final_html_m70797b4a.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7490/NCERT_Solution_Math_Chapter_7_final_html_42b7f736.gif

Let sin x + cos x = t 

⇒ (cos x − sin xdx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7490/NCERT_Solution_Math_Chapter_7_final_html_e9f2ce0.gif

Question 33:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7492/NCERT_Solution_Math_Chapter_7_final_html_70acf21f.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7492/NCERT_Solution_Math_Chapter_7_final_html_m1fecaf67.gif

Put cos x − sin x = t 

⇒ (−sin x − cos xdx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7492/NCERT_Solution_Math_Chapter_7_final_html_1c1841d8.gif

*Question 34:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7494/NCERT_Solution_Math_Chapter_7_final_html_m3cd2a5e1.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7494/NCERT_Solution_Math_Chapter_7_final_html_m441fc4b5.gif

*Question 35:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7497/NCERT_Solution_Math_Chapter_7_final_html_6ef9e05f.gif

Answer:

Let 1 + log x = t

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7497/NCERT_Solution_Math_Chapter_7_final_html_75d6f3b5.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7497/NCERT_Solution_Math_Chapter_7_final_html_m79272764.gif

*Question 36:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_663e2a5d.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_m7e821b61.gif

Let 

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_12ec54d0.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_m2f765048.gif

*Question 37:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_3d29e0a5.gif

Answer:

Let x4 = t

∴ 4x3 dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_m4c5a1b01.gif

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_m3d91c446.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_4d47be1c.gif

From (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_626f7efb.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_m776b9de8.gif

Question 38:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_3cbaf76f.gif

equals

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_21be7c9a.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_4e59dfbc.gif

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_m65c6fb2.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_m3b428ede.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_3b394c63.gif

Hence, the correct answer is D.

Question 39:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m52ea234.gif

equals

A.  B. 

C.  D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m3109d37e.gif

Hence, the correct answer is B.

Exercise-7.3

Question 1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7510/NCERT_Solution_Math_Chapter_7_final_html_158e4d81.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7510/NCERT_Solution_Math_Chapter_7_final_html_m25ad4f6f.gif

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7520/NCERT_Solution_Math_Chapter_7_final_html_m79d80b67.gif

Answer:

It is known that, 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7520/NCERT_Solution_Math_Chapter_7_final_html_4e6849bf.gif

Question 3:

cos 2x cos 4x cos 6x

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7523/NCERT_Solution_Math_Chapter_7_final_html_m453c4158.gif

Question 4:

sin3 (2x + 1)

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7525/NCERT_Solution_Math_Chapter_7_final_html_m634ea88b.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7525/NCERT_Solution_Math_Chapter_7_final_html_2257b9f2.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7525/NCERT_Solution_Math_Chapter_7_final_html_m5795a85f.gif

Question 5:

sin3 x cos3 x

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7527/NCERT_Solution_Math_Chapter_7_final_html_m73e75233.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7527/NCERT_Solution_Math_Chapter_7_final_html_69753c1b.gif

Question 6:

sin x sin 2x sin 3x

Answer:

It is known that, 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7529/NCERT_Solution_Math_Chapter_7_final_html_4b01ff05.gif

Question 7:

sin 4x sin 8x

Answer:

It is known that,

sin A . sin B = 12cos(A-B)-cos(A+B)

∴∫sin4x sin8x dx

=∫12cos4x-8x-cos4x+8xdx

=12∫cos-4x-cos12xdx

=12∫cos4x-cos12xdx

=12sin4x4-sin12x12+C

Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7533/NCERT_Solution_Math_Chapter_7_final_html_4d6e72c3.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/Kayako_FKY%2019079408.gif

Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7535/NCERT_Solution_Math_Chapter_7_final_html_7705fbfd.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7535/NCERT_Solution_Math_Chapter_7_final_html_m316e357f.gif

Question 10:

sin4 x

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7536/NCERT_Solution_Math_Chapter_7_final_html_m48648a1c.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7536/NCERT_Solution_Math_Chapter_7_final_html_m31f5f804.gif

Question 11:

cos4 2x

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7538/NCERT_Solution_Math_Chapter_7_final_html_m2f3cf2d6.gif

Question 12:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7540/NCERT_Solution_Math_Chapter_7_final_html_639207d4.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7540/NCERT_Solution_Math_Chapter_7_final_html_1b7154f9.gif

Question 13:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7543/NCERT_Solution_Math_Chapter_7_final_html_m3bf74025.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7543/NCERT_Solution_Math_Chapter_7_final_html_m3df62037.gif

Question 14:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7544/NCERT_Solution_Math_Chapter_7_final_html_m2f3df7c7.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7544/NCERT_Solution_Math_Chapter_7_final_html_m3ed05535.gif

Question 15:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7547/NCERT_Solution_Math_Chapter_7_final_html_61e38253.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7547/NCERT_Solution_Math_Chapter_7_final_html_m3ab061dc.gif

Question 16:

tan4x

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7549/NCERT_Solution_Math_Chapter_7_final_html_m781709fe.gif

From equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7549/NCERT_Solution_Math_Chapter_7_final_html_m15daa35b.gif

Question 17:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7552/NCERT_Solution_Math_Chapter_7_final_html_390003e3.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7552/NCERT_Solution_Math_Chapter_7_final_html_m4da9ac1c.gif

Question 18:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7554/NCERT_Solution_Math_Chapter_7_final_html_m4c746474.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7554/NCERT_Solution_Math_Chapter_7_final_html_2529779e.gif

Question 19:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7557/NCERT_Solution_Math_Chapter_7_final_html_m473f7a6d.gif

Answer:

eWwNOr qqtaFh6M Mbvl1nuTO41ambz IBiB6VHA4duAZkS hkrdXFYgHwWfgU AYvo1huBKxxTfVY5iI bYMYv0cNwaxJgMpW 5ndlk1kZf Oz99FO5whdL pFbQmVtugY1FU
EBGuJQ96ip3lCx7w6Sd0HYfRt2NjTUOOlQSkYdtAowAdkY8CrsoHSR57AMUvRuOzbib4Y8pX43GUuUwzsWD 50qHypx4lJ8RrC8aUm9mIkwUg5fORq
E YWqODn5 Iu9M124jPwws L78nPEXpLiYBA6bOksEW59BebuxluJ56W4JwzB5mMqC0anZdE
https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/zafar1111.png

*Question 20:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7559/NCERT_Solution_Math_Chapter_7_final_html_3ac37af4.gif

Answer:

RjPux3JIfV2S5KJ5ebpVpGB2MnQ TlAJU2rsM BUNjvcCzNJaCLdiH7imCHnOzJKTQeIW4eR hCaclh1zfITs Rd3m5ZUurZ9momiSu7qRte sNepAue5d4Oy8 3lcBwbU1Tu50
Zfe ZJNaXQuKuCpPxVYqePdizMUl6PxtjccReOGCTEBKIjMrPv18 3

*Question 21:

sin−1 (cos x)

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_5ca185ec.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_34c59b7.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_67f31dd1.gif

Substituting in equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_62bbad50.gif

*Question 22:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7564/NCERT_Solution_Math_Chapter_7_final_html_m299efddf.gif

Answer:

bFmVbDWdT6HwmSQe585H9EWyGwonEp5aYoDlu 4Q gf6GFq6OzRCu0tuKmWG tB 4xE3JeyE OknFY23VqrpKQxq 6nLewXg6DrehQIix5o2nBAkDPAeX du1lEpmKidv9VMNpA

*Question 23:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7566/NCERT_Solution_Math_Chapter_7_final_html_m356fefc4.gif is equal to

A.  tan x + cot x + C

B.  tan x + cosec x + C

C.  −tan x + cot x + C

D.  tan x + sec x + C

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7566/NCERT_Solution_Math_Chapter_7_final_html_m5794483e.gif

Hence, the correct answer is A.

*Question 24:

 equals

A. 

B. 

C. 

D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7568/NCERT_Solution_Math_Chapter_7_final_html_m46251d8.gif

Let (e^x).x = t

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7568/NCERT_Solution_Math_Chapter_7_final_html_m410bc16d.gif

Hence, the correct answer is B.

Exercise-7.4

Question 1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7570/NCERT_Solution_Math_Chapter_7_final_html_m6d4a512a.gif

Answer:

Let x3 = t

∴ 3x2 dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7570/NCERT_Solution_Math_Chapter_7_final_html_6c319a08.gif

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7572/NCERT_Solution_Math_Chapter_7_final_html_m6eadee56.gif

Answer:

Let 2x = t

∴ 2dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7572/NCERT_Solution_Math_Chapter_7_final_html_2bb9ddb6.gif

Question 3:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7574/NCERT_Solution_Math_Chapter_7_final_html_m4f26274f.gif

Answer:

Let 2 − t

⇒ −dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7574/NCERT_Solution_Math_Chapter_7_final_html_m311edd7e.gif

Question 4:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7576/NCERT_Solution_Math_Chapter_7_final_html_12ee8f5b.gif

Answer:

Let 5x = t

∴ 5dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7576/NCERT_Solution_Math_Chapter_7_final_html_4152a2c9.gif

Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7577/NCERT_Solution_Math_Chapter_7_final_html_1313bb5.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7577/NCERT_Solution_Math_Chapter_7_final_html_521d393d.gif

Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7579/NCERT_Solution_Math_Chapter_7_final_html_686a690e.gif

Answer:

Let x3 = t

∴ 3x2 dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7579/NCERT_Solution_Math_Chapter_7_final_html_687cb876.gif

Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7581/NCERT_Solution_Math_Chapter_7_final_html_m693dd56.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7581/NCERT_Solution_Math_Chapter_7_final_html_m6e073184.gif

From (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7581/NCERT_Solution_Math_Chapter_7_final_html_m5bd0fa9b.gif

Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7583/NCERT_Solution_Math_Chapter_7_final_html_1f129e0a.gif

Answer:

Let x3 = t

⇒ 3x2 dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7583/NCERT_Solution_Math_Chapter_7_final_html_m163da311.gif

Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7584/NCERT_Solution_Math_Chapter_7_final_html_m46808fa3.gif

Answer:

Let tan x = t

∴ sec2x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7584/NCERT_Solution_Math_Chapter_7_final_html_654e25c6.gif

Question 10:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7586/NCERT_Solution_Math_Chapter_7_final_html_m94a3620.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7586/NCERT_Solution_Math_Chapter_7_final_html_m518e3476.gif

Question 11:

Qsahkaosp632Gp F0NSF41wVQ1dAzpbwTMW UzgVaoAphCSxK0s3b0BM0XPdzP9gAAeaqiSMAQl

Answer:

GtCkBgboTYGHXo2IQuUwMOvwABUfiwaUG1IwulV4CJ kYUB8W33pyk KQwbggmhwhHcfFo 0CdpbDDgYK6e2NGosJeSSq9RoyJn3bJ9bcoEGZBTgRUQPr nFeezRIpDYlv7UUtg

=

 Let (3x+1)=t

∴3 dx = dt

=(1/3)

99TQfz4FRW1sKwRi3Ta3BmW2yP6OgA5tMctd0UDxnyNX8lB6bIr56R5CJsFDahkgwbZAGXo is7xAtxHJ8wtzBN 8thsiwGWBSeQQ1gZUSnRMMN6ifvBut1G6mRj5LeOPo8zZzU
ALZm2vY28574cfXarPOGP1Pg3SLwpV7FFHCUAZOj6irLKAX9ztIsvujcS5 j6GFGtAf1zteIML1uqEn2unan O04Ncw8efHfwF3G82vuHy03mqQcHjggRJ7qMuLUvcrviYybZk

Question 12:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7589/NCERT_Solution_Math_Chapter_7_final_html_168db442.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7589/NCERT_Solution_Math_Chapter_7_final_html_10a5934e.gif

Question 13:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7591/NCERT_Solution_Math_Chapter_7_final_html_62650cce.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7591/NCERT_Solution_Math_Chapter_7_final_html_m7b6ec913.gif

Question 14:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7593/NCERT_Solution_Math_Chapter_7_final_html_62f95dc1.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7593/NCERT_Solution_Math_Chapter_7_final_html_m60221a62.gif

Question 15:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7596/NCERT_Solution_Math_Chapter_7_final_html_m67d6b929.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7596/NCERT_Solution_Math_Chapter_7_final_html_m1d0caab4.gif

Question 16:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7598/NCERT_Solution_Math_Chapter_7_final_html_664b5d67.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7598/NCERT_Solution_Math_Chapter_7_final_html_m1b1241f3.gif

Equating the coefficients of x and constant term on both sides, we obtain

4A = 4 ⇒ A = 1

A + B = 1 ⇒ B = 0

Let 2x2 + x − 3 = t

∴ (4x + 1) dx dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7598/NCERT_Solution_Math_Chapter_7_final_html_2a4a7359.gif

Question 17:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_39c73fad.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_b662a70.gif

Equating the coefficients of x and constant term on both sides, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_m668374ed.gif

From (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_1ddb53d5.gif

From equation (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_7731a74.gif

Question 18:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m59186f15.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m72fdaa74.gif

Equating the coefficient of x and constant term on both sides, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m1c9032eb.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m2bfaa834.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_1ab063c8.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m192518e9.gif

Substituting equations (2) and (3) in equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m4ebced4d.gif

Question 19:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m9c52b48.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_6a0bbc55.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m5ae3a8ee.gif

Equating the coefficients of x and constant term, we obtain

2A = 6 ⇒ A = 3

−9A + B = 7 ⇒ B = 34

∴ 6x + 7 = 3 (2x − 9) + 34

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m597fb40f.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m55743fdf.gif

Substituting equations (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m151359b3.gif

Question 20:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_3a973c5a.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_m1e4b7ad4.gif

Equating the coefficients of x and constant term on both sides, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_m68eba682.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_17bfe748.gif

Using equations (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_m70c40c54.gif

Question 21:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_4ba941c2.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_m5aaab5ed.gif

Let x2 + 2x +3 = t

⇒ (2x + 2) dx =dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_1869b74b.gif

Using equations (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_m53d4ecad.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_m67495a4.gif

*Question 22:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_m36d7c819.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_7848b5f8.gif

Equating the coefficients of x and constant term on both sides, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_4b12c12d.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_m713d5001.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_68d718b3.gif

Substituting (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_6aab90c.gif

*Question 23:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_17e7b92.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_445b6746.gif

Equating the coefficients of x and constant term, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_m58eeddc7.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_m224dadf8.gif

Using equations (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_m479038b2.gif

*Question 24:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7629/NCERT_Solution_Math_Chapter_7_final_html_492a94c1.gif

equals

A.  x tan−1 (x + 1) + C

B.  tan− 1 (x + 1) + C

C.  (x + 1) tan−1 x + C

D.  tan−1 x + C

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7629/NCERT_Solution_Math_Chapter_7_final_html_m59b21429.gif

Hence, the correct answer is B.

*Question 25:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7631/NCERT_Solution_Math_Chapter_7_final_html_m55ee7506.gifequals

A. 

B. 

C. 

D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7631/NCERT_Solution_Math_Chapter_7_final_html_m563bf5a5.gif

Hence, the correct answer is B.

Exercise-7.5

Question 1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7634/NCERT_Solution_Math_Chapter_7_final_html_m55126bc.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7634/NCERT_Solution_Math_Chapter_7_final_html_mdc04adc.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7634/NCERT_Solution_Math_Chapter_7_final_html_m16112759.gif

Equating the coefficients of x and constant term, we obtain

A + = 1

2A + B = 0

On solving, we obtain

A = −1 and B = 2

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7634/NCERT_Solution_Math_Chapter_7_final_html_m4848e6b0.gif

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7636/NCERT_Solution_Math_Chapter_7_final_html_m68b00263.gif

Answer:

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7636/NCERT_Solution_Math_Chapter_7_final_html_m28ec0239.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7636/NCERT_Solution_Math_Chapter_7_final_html_m28783f7d.gif

Equating the coefficients of x and constant term, we obtain

A + B = 0

−3A + 3B = 1

On solving, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7636/NCERT_Solution_Math_Chapter_7_final_html_535f7aa7.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7636/NCERT_Solution_Math_Chapter_7_final_html_m3d20aa7e.gif

Question 3:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7637/NCERT_Solution_Math_Chapter_7_final_html_47ceee9b.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7637/NCERT_Solution_Math_Chapter_7_final_html_m4f1e46cd.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7637/NCERT_Solution_Math_Chapter_7_final_html_26acb356.gif

Substituting x = 1, 2, and 3 respectively in equation (1), we obtain

A = 1, B = −5, and C = 4

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7637/NCERT_Solution_Math_Chapter_7_final_html_m4450a170.gif

Question 4:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7638/NCERT_Solution_Math_Chapter_7_final_html_m604be94a.gif

Answer:

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7638/NCERT_Solution_Math_Chapter_7_final_html_71020d43.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7638/NCERT_Solution_Math_Chapter_7_final_html_m1762d17b.gif

Substituting x = 1, 2, and 3 respectively in equation (1), we obtain  and

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7638/NCERT_Solution_Math_Chapter_7_final_html_m3e97765a.gif

Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7640/NCERT_Solution_Math_Chapter_7_final_html_67c72223.gif

Answer:

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7640/NCERT_Solution_Math_Chapter_7_final_html_md82450c.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7640/NCERT_Solution_Math_Chapter_7_final_html_51425d46.gif

Substituting x = −1 and −2 in equation (1), we obtain

A = −2 and B = 4

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7640/NCERT_Solution_Math_Chapter_7_final_html_44d773ed.gif

Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_m283b87e2.gif

Answer:

It can be seen that the given integrand is not a proper fraction.

Therefore, on dividing (1 − x2) by x(1 − 2x), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_m6d6cb64.gif

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_37abe9b6.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_7abf94a2.gif

Substituting x = 0 and in equation (1), we obtain

= 2 and B = 3

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_m633e2bc7.gif

Substituting in equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_m5ef621d.gif

Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7646/NCERT_Solution_Math_Chapter_7_final_html_4c300254.gif

Answer:

Let 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7646/NCERT_Solution_Math_Chapter_7_final_html_m4f316a86.gif

Equating the coefficients of x2x, and constant term, we obtain

A + C = 0

A + B = 1

B + C = 0

On solving these equations, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7646/NCERT_Solution_Math_Chapter_7_final_html_1512cd64.gif

From equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7646/NCERT_Solution_Math_Chapter_7_final_html_m657b90b0.gif

Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_33a31e6c.gif

Answer:

Let 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_m50fcae82.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_4f0f5909.gif

Substituting x = 1, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_m16a77e49.gif

Equating the coefficients of x2 and constant term, we obtain

A + C = 0

−2A + 2B + C = 0

On solving, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_m620277b2.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_58e6a15b.gif

Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_15b962ca.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_4a16e570.gif

Let 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_33f40317.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_m7cd0d550.gif

Substituting x = 1 in equation (1), we obtain

B = 4

Equating the coefficients of x2 and x, we obtain

A + C = 0

B − 2C = 3

On solving, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_m643a66d1.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_m4e34711f.gif

Question 10:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_7dfa52ef.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_mcf89333.gif

Let 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_3f730617.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_48a2f765.gif

Equating the coefficients of x2 and x, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_m4ee3b74.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_6a680da4.gif

Question 11:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_meef6cee.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_7f58a1eb.gif

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_mad7d3e3.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_m3492ed42.gif

Substituting = −1, −2, and 2 respectively in equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_5b116a13.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_m4654c7c5.gif

Question 12:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_m7e9bdf6f.gif

Answer:

It can be seen that the given integrand is not a proper fraction.

Therefore, on dividing (x3 + x + 1) by x2 − 1, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_m70e87107.gif

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_6649a66d.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_m3d304e42.gif

Substituting = 1 and −1 in equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_m117d1cc.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_9b89357.gif

Question 13:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7657/NCERT_Solution_Math_Chapter_7_final_html_1b0b274d.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7657/NCERT_Solution_Math_Chapter_7_final_html_31515503.gif

Equating the coefficient of x2x, and constant term, we obtain

A − B = 0

B − C = 0

A + C = 2

On solving these equations, we obtain

A = 1, B = 1, and C = 1

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7657/NCERT_Solution_Math_Chapter_7_final_html_m6a802ca2.gif

Question 14:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7659/NCERT_Solution_Math_Chapter_7_final_html_1d0293db.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7659/NCERT_Solution_Math_Chapter_7_final_html_m151d55ff.gif

Equating the coefficient of x and constant term, we obtain

A = 3

2A + = −1 ⇒ B = −7

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7659/NCERT_Solution_Math_Chapter_7_final_html_m46af344a.gif

Question 15:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_m63e2a732.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_m23dc4fc1.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_1b491979.gif

Equating the coefficient of x3x2, x, and constant term, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_m712c361.gif

On solving these equations, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_m4d8f1b57.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_a36f335.gif

Question 16:

 [Hint: multiply numerator and denominator by  and put xn = t]

Answer:

dZ B3zvniiu fx9X3qja7Z9EZ7USp2TgTZ9wmVPN1kaSDTl0K6 HMn k OFKyqlE7 ON8D9 q1LMn AUMIq8lN v8dmHt7NfwX3KhYE46Jnvs Kk28X iaiFYkhY5WS4kVSQpQg

Multiplying numerator and denominator by , we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7663/NCERT_Solution_Math_Chapter_7_final_html_m202e0f7f.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7663/NCERT_Solution_Math_Chapter_7_final_html_m25513d6a.gif

Substituting t = 0, −1 in equation (1), we obtain

A = 1 and B = −1

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7663/NCERT_Solution_Math_Chapter_7_final_html_m1437c43a.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7663/NCERT_Solution_Math_Chapter_7_final_html_m3402fc01.gif

Question 17:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7665/NCERT_Solution_Math_Chapter_7_final_html_6343aea2.gif

[Hint: Put sin x = t]

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7665/NCERT_Solution_Math_Chapter_7_final_html_40e59867.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7665/NCERT_Solution_Math_Chapter_7_final_html_1c3aabf6.gif

Substituting t = 2 and then t = 1 in equation (1), we obtain

A = 1 and B = −1

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7665/NCERT_Solution_Math_Chapter_7_final_html_1bd280c7.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7665/NCERT_Solution_Math_Chapter_7_final_html_m62760a7e.gif

Question 18:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7668/NCERT_Solution_Math_Chapter_7_final_html_63cc9e04.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7668/NCERT_Solution_Math_Chapter_7_final_html_67db95a0.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7668/NCERT_Solution_Math_Chapter_7_final_html_14356613.gif

Equating the coefficients of x3x2x, and constant term, we obtain

A + C = 0

B + D = 4

4A + 3C = 0

4B + 3D = 10

On solving these equations, we obtain

A = 0, B = −2, C = 0, and D = 6

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7668/NCERT_Solution_Math_Chapter_7_final_html_473e50ca.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7668/NCERT_Solution_Math_Chapter_7_final_html_591d2419.gif

*Question 19:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_29c714c8.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_29c714c8.gif

Let x2 = t 

⇒ 2x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_709f9dff.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_m740aa2df.gif

Substituting = −3 and = −1 in equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_693a6469.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_m68da9ac5.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_2c43a007.gif

*Question 20:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_12c7d30a.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_12c7d30a.gif

Multiplying numerator and denominator by x3, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_36624ea7.gif

Let x4 = t 

⇒ 4x3dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_6efe54f8.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_m40c17f5.gif

Substituting t = 0 and 1 in (1), we obtain

A = −1 and B = 1

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_5a5b384f.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_m4c782b7a.gif

*Question 21:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_m15581dd.gif

[Hint: Put ex = t]

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_m15581dd.gif

Let ex = 

⇒ ex dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_m5058b23.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_16b2ac81.gif

Substituting t = 1 and t = 0 in equation (1), we obtain

A = −1 and B = 1

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_237dc92f.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_m4e47628f.gif

*Question 22:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7673/NCERT_Solution_Math_Chapter_7_final_html_m3f6b4c23.gif

A.  B. 

C.  D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7673/NCERT_Solution_Math_Chapter_7_final_html_m1a39d92a.gif

Substituting x = 1 and 2 in (1), we obtain

A = −1 and B = 2

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7673/NCERT_Solution_Math_Chapter_7_final_html_m1da97f65.gif

Hence, the correct answer is B.

*Question 23:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7674/NCERT_Solution_Math_Chapter_7_final_html_3ee528b8.gif

A.  B. 

C.  D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7674/NCERT_Solution_Math_Chapter_7_final_html_1c4b44d7.gif

Equating the coefficients of x2x, and constant term, we obtain

A + B = 0

C = 0

A = 1

On solving these equations, we obtain

= 1, B = −1, and C = 0

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7674/NCERT_Solution_Math_Chapter_7_final_html_6b082199.gif

Hence, the correct answer is A.

Exercise-7.6

Question 1:

x sin x

Answer:

Let 

Taking x as first function and sin x as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7675/NCERT_Solution_Math_Chapter_7_final_html_m5b52456b.gif

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7676/NCERT_Solution_Math_Chapter_7_final_html_m565b8df6.gif

Answer:

Let 

Taking x as first function and sin 3x as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7676/NCERT_Solution_Math_Chapter_7_final_html_m214c7ae3.gif

Question 3:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_45da2d8.gif

Answer:

Let 

Taking x2 as first function and ex as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_m1406435d.gif

Again, integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_243d97bb.gif

Question 4:

x logx

Answer:

Let 

Taking log x as first function and x as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7678/NCERT_Solution_Math_Chapter_7_final_html_m5b95e34d.gif

Question 5:

x log 2x

Answer:

Let 

Taking log 2x as first function and x as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7679/NCERT_Solution_Math_Chapter_7_final_html_m50a12fc3.gif

Question 6:

xlog x

Answer:

Let 

Taking log x as first function and x2 as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7680/NCERT_Solution_Math_Chapter_7_final_html_126181e4.gif

Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7681/NCERT_Solution_Math_Chapter_7_final_html_m70b6b8fb.gif

Answer:

Let 

Taking as first function and x as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7681/NCERT_Solution_Math_Chapter_7_final_html_m3a459646.gif

Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7682/NCERT_Solution_Math_Chapter_7_final_html_11d5bd80.gif

Answer:

Let 

Taking as first function and x as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7682/NCERT_Solution_Math_Chapter_7_final_html_m19e79690.gif

Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7683/NCERT_Solution_Math_Chapter_7_final_html_5ad5a2f.gif

Answer:

Let 

Taking cos−1 x as first function and x as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7683/NCERT_Solution_Math_Chapter_7_final_html_1991c6d7.gif

Question 10:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7684/NCERT_Solution_Math_Chapter_7_final_html_2aa59a77.gif

Answer:

Let 

Taking as first function and 1 as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7684/NCERT_Solution_Math_Chapter_7_final_html_783e653d.gif

Question 11:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_m7beafb64.gif

Answer:

Let 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_m54c945f9.gif

Taking https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_m47731349.gif as first function and as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_56d7e816.gif

Question 12:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7687/NCERT_Solution_Math_Chapter_7_final_html_4e99fa3a.gif

Answer:

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7687/NCERT_Solution_Math_Chapter_7_final_html_5977bb13.gif

Taking x as first function and sec2x as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7687/NCERT_Solution_Math_Chapter_7_final_html_m384594e7.gif

Question 13:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7688/NCERT_Solution_Math_Chapter_7_final_html_m1a8b5181.gif

Answer:

Let 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7688/NCERT_Solution_Math_Chapter_7_final_html_226da9cf.gif

Taking  as first function and 1 as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7688/NCERT_Solution_Math_Chapter_7_final_html_m4bf69e04.gif

Question 14:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7689/NCERT_Solution_Math_Chapter_7_final_html_541373dc.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7689/NCERT_Solution_Math_Chapter_7_final_html_m783b9696.gif

Taking  as first function and x as second function and integrating by parts, we obtain

dd8vm EZ9fyozYVG jYxu cjclBCUg1L0HFdUj5fCX0qZWbwhj6 PCvV0dwxMoEvZ5rvg0Q8JsZE1sgFNsuqvn3ARdnx jn4hijDSth39dnqb8 IOTsWcjemgiSXREP QQ MLQ
BcgcohYJhYo6LkNA0cn0nuWoM ZhKWyDBKM7ASDT9vBGSDJt0QbVbU2ZOL4nnQnxpmAWkaXpxD89d7jBGef3Lgr R5CMYFoLZHF02A0m4iCn1f JdOTQSFkMVrsPmOUemyMBAU
00Iz7C4XfF d0cyu1hatyX

Again, integrating by parts, we obtain

cs1jfIdPxZURA1Y3CnAPEhsKvMq5ve5Fex4JRBCZjCiziJhjcsgMFYSVxpKStngaBug agAfMea3AITIBdb1CcJYM DZbkLB1mkslriqOI9BPthCqx OkDqJq7RmXvSeTefRHVA
2qWKHNewVx64DNJWRqgjuWlJM KiadPw02ToAV0RoQeOv
M2z13D8lpUCXr2egwNRLNuAAD7j3K7ZNU4 qbdNggyxNgt twxLKLSBvamhn6LkdiOKQq7x06xSGrkqZ yvppipc2WZNRPAa67prGnIEHV9xOdXXRNPta9D0W5NWJ57vS mDFS0

Question 15:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m7e288453.gif

Answer:

Let 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m7d7d9059.gif

Let I = I1 + I2 … (1)

Where,

  and

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_508ac40a.gif

Taking log x as first function and xas second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_64161ae9.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m3afbd7eb.gif

Taking log x as first function and 1 as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m33301466.gif

Using equations (2) and (3) in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_32c47506.gif

Question 16:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_m16dbffa1.gif

Answer:

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_m3fc69b14.gif

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_8ec6dc1.gif

⇒ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_2d6339dd.gif

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_264157a1.gif

It is known that

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_ma0bf94e.gif

Question 17:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_41e8a549.gif

Answer:

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_f2746bb.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_356f9e57.gif

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_4bcecbea.gif ⇒ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_m311bf58a.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_m692fc956.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_m234a6149.gif

Question 18:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_2d60c21c.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_1b8fb60a.gif

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_23c6706d.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_23e4f06.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif

From equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_m483f6657.gif

Question 19:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_425f49ab.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_509a61e0.gif

Also, let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_m47d9edac.gif 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_m123a8371.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_m7a0cc9fc.gif

*Question 20:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m5d04b491.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m1ffb9b2d.gif

Let

 https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_2cc6670b.gif ⇒ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m4e016495.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m3379bca.gif

*Question 21:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m198871a2.gif

Answer:

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m349755a1.gif

Integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m72845c80.gif

Again, integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m624ee5cc.gif

*Question 22:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_5c4b1fa6.gif

Answer:

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_2effc955.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_m5b0e6d85.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_2c176593.gif

= 2θ

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_1772d35e.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_m2a375b8f.gif

Integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_m2f12cf1c.gif

*Question 23:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_36e73ee8.gif

 equals

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_m2073483f.gif

Answer:

Let 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_56dd90a3.gif

Also, let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_770f1019.gif

⇒ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_27cd530.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_m3e91749d.gif

Hence, the correct answer is A.

*Question 24:

 equals

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_42378067.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_46f2088a.gif

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_594a5f7.gif

Also, let https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_m1536fdf4.gif 

⇒ https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_m7b826a5e.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_20bec91b.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_31dddd75.gif

Hence, the correct answer is B.

Exercise-7.7

Question 1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7716/NCERT_Solution_Math_Chapter_7_final_html_m1170ca09.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_14_13_39/NCERT_Solution_Math_Chapter_7_final_html_55b76281_1748512521494652501.png
https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_14_13_39/NCERT_Solution_Math_Chapter_7_final_html_55b76281_4482949128659630918.png
https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_14_13_39/NCERT_Solution_Math_Chapter_7_final_html_55b76281_6819830928611292475.png
https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_14_13_39/NCERT_Solution_Math_Chapter_7_final_html_55b76281_5766645889198152699.png
https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_14_13_39/NCERT_Solution_Math_Chapter_7_final_html_55b76281_2274410893609283042.png

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7719/NCERT_Solution_Math_Chapter_7_final_html_m524978b7.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7719/NCERT_Solution_Math_Chapter_7_final_html_m25d68940.gif

Question 3:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7721/NCERT_Solution_Math_Chapter_7_final_html_1078b016.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7721/NCERT_Solution_Math_Chapter_7_final_html_12ee79a6.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7721/NCERT_Solution_Math_Chapter_7_final_html_3dc605c1.gif

Question 4:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7722/NCERT_Solution_Math_Chapter_7_final_html_m2833efcd.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7722/NCERT_Solution_Math_Chapter_7_final_html_6c3bdbca.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7722/NCERT_Solution_Math_Chapter_7_final_html_m5cf48318.gif

Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7724/NCERT_Solution_Math_Chapter_7_final_html_4046b156.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7724/NCERT_Solution_Math_Chapter_7_final_html_3fe6d1f5.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7724/NCERT_Solution_Math_Chapter_7_final_html_49e1f051.gif

Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7725/NCERT_Solution_Math_Chapter_7_final_html_m54dece65.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7725/NCERT_Solution_Math_Chapter_7_final_html_7cc40572.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7725/NCERT_Solution_Math_Chapter_7_final_html_482a9a24.gif

Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7727/NCERT_Solution_Math_Chapter_7_final_html_m29dd6f5d.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7727/NCERT_Solution_Math_Chapter_7_final_html_m2460c660.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7727/NCERT_Solution_Math_Chapter_7_final_html_m5fffe454.gif

*Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7729/NCERT_Solution_Math_Chapter_7_final_html_m19083a3b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7729/NCERT_Solution_Math_Chapter_7_final_html_m3e6f3b43.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7729/NCERT_Solution_Math_Chapter_7_final_html_m16dfab03.gif

*Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7731/NCERT_Solution_Math_Chapter_7_final_html_191c58cb.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7731/NCERT_Solution_Math_Chapter_7_final_html_m3f7229fc.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7731/NCERT_Solution_Math_Chapter_7_final_html_m5f522660.gif

*Question 10:

is equal to

A.  B. 

C.  D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7732/NCERT_Solution_Math_Chapter_7_final_html_m6520b32b.gif

Hence, the correct answer is A.

*Question 11:

is equal to

A. 

B. 

C. 

D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7733/NCERT_Solution_Math_Chapter_7_final_html_56279587.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7733/NCERT_Solution_Math_Chapter_7_final_html_m32d28a69.gif

Hence, the correct answer is D.

Exercise-7.8

Question 1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7734/NCERT_Solution_Math_Chapter_7_final_html_69e5f3b9.gif

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7734/NCERT_Solution_Math_Chapter_7_final_html_me7fec4d.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7734/NCERT_Solution_Math_Chapter_7_final_html_72e7930a.gif

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7735/NCERT_Solution_Math_Chapter_7_final_html_m7ea19f1b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7735/NCERT_Solution_Math_Chapter_7_final_html_5fe9d3a4.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7735/NCERT_Solution_Math_Chapter_7_final_html_m405877a1.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7735/NCERT_Solution_Math_Chapter_7_final_html_446ceb1.gif

Question 3:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7736/NCERT_Solution_Math_Chapter_7_final_html_m360250ef.gif

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7736/NCERT_Solution_Math_Chapter_7_final_html_m4f945515.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7736/NCERT_Solution_Math_Chapter_7_final_html_2aa04a39.gif

Question 4:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_m220fbce.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_m2879635d.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_m383850e.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_m6bd88959.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_5952197b.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_47ebd74.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_51a9ceb8.gif

From equations (2) and (3), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_1bb6734a.gif

Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7739/NCERT_Solution_Math_Chapter_7_final_html_m1cc03f8e.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7739/NCERT_Solution_Math_Chapter_7_final_html_57b2e480.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7739/NCERT_Solution_Math_Chapter_7_final_html_m4ddc80d9.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7739/NCERT_Solution_Math_Chapter_7_final_html_7c5270a4.gif

Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7741/NCERT_Solution_Math_Chapter_7_final_html_2690150b.gif

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7741/NCERT_Solution_Math_Chapter_7_final_html_m23e187a5.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7741/NCERT_Solution_Math_Chapter_7_final_html_164cb7ca.gif

Exercise-7.9

Question 1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7742/NCERT_Solution_Math_Chapter_7_final_html_m1067703e.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7742/NCERT_Solution_Math_Chapter_7_final_html_39473b81.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7742/NCERT_Solution_Math_Chapter_7_final_html_m5b213be6.gif

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7743/NCERT_Solution_Math_Chapter_7_final_html_480f9557.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7743/NCERT_Solution_Math_Chapter_7_final_html_51690e6.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7743/NCERT_Solution_Math_Chapter_7_final_html_303663d0.gif

Question 3:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7745/NCERT_Solution_Math_Chapter_7_final_html_m70152f9b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7745/NCERT_Solution_Math_Chapter_7_final_html_m20e8f6a9.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7745/NCERT_Solution_Math_Chapter_7_final_html_m35238a09.gif

Question 4:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7746/NCERT_Solution_Math_Chapter_7_final_html_4ba6d54a.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7746/NCERT_Solution_Math_Chapter_7_final_html_m2e6c9f5f.gif

By second fundamental theorem of calculus, we obtain

r3lVGL4WbszQWsYgyA8ninnG4yW1bkaQ6xtouFXdKmUtRfTXXMErMFdQMSTMyTgHRGcmEdlLpsRlh5nIZWA154HqMNjxbl
eiyVWIJ50UuUfOQqiZuzkhWTe5vLF2D3U8iouUa6gdBAFHqHVug4pecL hHhvf9uItIwBV7bzBcnovbrPgWmNYMOV LQOPQj6BzgmZTEYTZSlwSMkUeOYSlKUG1Coc0nHBGHFSQ
awEJawgzepUHvhaPqO27370MfXOzHK9RXHNHCG TnkVx HVMnRQTqqtBy990uuKC2ncGd 13
CdcJCSomzuM jnYMU NVjn4Go3fIE4ziqmsQ8Bk0bPItEAWngKGu8k55KtQw3xyjyqZH afkKwU9tVK3o4FUSHO6p57lBGeY7n Od NvWNjRLZBpDnwfa6tX1FIehXONtbi s8

Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7747/Items%20179_html_1c52a724.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7747/Items%20179_html_608a452e.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7747/Items%20179_html_m3cd3c92d.gif

Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7749/NCERT_Solution_Math_Chapter_7_final_html_426fe2e4.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7749/NCERT_Solution_Math_Chapter_7_final_html_m5c2222aa.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7749/NCERT_Solution_Math_Chapter_7_final_html_m6026a9a1.gif

Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7751/NCERT_Solution_Math_Chapter_7_final_html_m38d7921f.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7751/NCERT_Solution_Math_Chapter_7_final_html_m4d2e837b.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7751/NCERT_Solution_Math_Chapter_7_final_html_m44340047.gif

Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7752/NCERT_Solution_Math_Chapter_7_final_html_m1498ff3f.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7752/NCERT_Solution_Math_Chapter_7_final_html_m6da61a7.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7752/NCERT_Solution_Math_Chapter_7_final_html_m7c581a2d.gif

Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7754/NCERT_Solution_Math_Chapter_7_final_html_m166e5b62.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7754/NCERT_Solution_Math_Chapter_7_final_html_m2b080397.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7754/NCERT_Solution_Math_Chapter_7_final_html_14d81098.gif

Question 10:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7756/NCERT_Solution_Math_Chapter_7_final_html_m81460a9.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7756/NCERT_Solution_Math_Chapter_7_final_html_m4f2ac16.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7756/NCERT_Solution_Math_Chapter_7_final_html_m2c4074b2.gif

Question 11:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7760/NCERT_Solution_Math_Chapter_7_final_html_682faf17.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7760/NCERT_Solution_Math_Chapter_7_final_html_61f2f3f1.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7760/NCERT_Solution_Math_Chapter_7_final_html_m2f2f6fbd.gif

Question 12:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7762/Item%20186_html_407b64a3.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7762/Item%20186_html_7c21c7f5.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7762/Item%20186_html_m19730f06.gif

Question 13:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7763/NCERT_Solution_Math_Chapter_7_final_html_79211595.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7763/NCERT_Solution_Math_Chapter_7_final_html_m9bd612c.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7763/NCERT_Solution_Math_Chapter_7_final_html_m45040b1.gif

Question 14:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7764/NCERT_Solution_Math_Chapter_7_final_html_1c0733f6.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7764/NCERT_Solution_Math_Chapter_7_final_html_7b54546e.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7764/NCERT_Solution_Math_Chapter_7_final_html_mebefe21.gif

Question 15:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7765/NCERT_Solution_Math_Chapter_7_final_html_6d91cb30.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7765/NCERT_Solution_Math_Chapter_7_final_html_m6ca16809.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7765/NCERT_Solution_Math_Chapter_7_final_html_m2160b3b9.gif

Question 16:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_1a209d0c.gif

Answer:

Let

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_ma123791.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_m3e58e905.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_2453d8c5.gif

Equating the coefficients of x and constant term, we obtain

A = 10 and B = −25

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_91e9a9d.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_90155e9.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_m9471e1f.gif

Substituting the value of I1 in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_m6af71c41.gif

Question 17:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7767/NCERT_Solution_Math_Chapter_7_final_html_62748c4f.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7767/NCERT_Solution_Math_Chapter_7_final_html_799ad7cb.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7767/NCERT_Solution_Math_Chapter_7_final_html_m160b5348.gif

Question 18:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7769/Item%20192_html_m1a378172.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7769/Item%20192_html_3225002c.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7769/Item%20192_html_m58ae84a8.gif

*Question 19:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7771/Item%20193_html_24e76ff5.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7771/Item%20193_html_6ea94b8.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7771/Item%20193_html_17842141.gif

*Question 20:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7774/Item%20194_html_m53e235e1.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7774/Item%20194_html_662f5233.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7774/Item%20194_html_1dd66a6f.gif

*Question 21:

equals

A.  B.  C.  D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7777/NCERT_Solution_Math_Chapter_7_final_html_m4a2a6a82.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7777/NCERT_Solution_Math_Chapter_7_final_html_m6107f62f.gif

Hence, the correct answer is D.

*Question 22:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7780/Item%20196_html_m45705453.gifequals

A.  B.  C.  D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7780/Item%20196_html_39a413bc.gif

By second fundamental theorem of calculus, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7780/Item%20196_html_3ccb1065.gif

Hence, the correct answer is C.

Exercise-7.10

Question 1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7784/NCERT_Solution_Math_Chapter_7_final_html_6d4f200d.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7784/NCERT_Solution_Math_Chapter_7_final_html_26c4315c.gif

When x = 0, t = 1 and when x = 1, t = 2

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7784/NCERT_Solution_Math_Chapter_7_final_html_mb530b8a.gif

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_m4c0d7ecb.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_m4f40448a.gif

Also, let 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_e432d87.gif

Question 3:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_m37dfca2b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_m5f5ba4a2.gif

Also, let x = tanθ 

⇒ dx = sec2θ dθ

When x = 0, θ = 0 and when = 1, 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_7416ad9b.gif

Takingas first function and sec2θ as second function and integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_m1696ceaa.gif

Question 4:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7792/NCERT_Solution_Math_Chapter_7_final_html_m6b4ab1ea.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7792/NCERT_Solution_Math_Chapter_7_final_html_m4fa51e4.gif

Let + 2 = t2 

⇒ dx = 2tdt

When x = 0,  and when = 2, = 2

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7792/NCERT_Solution_Math_Chapter_7_final_html_671afa1.gif

Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_m21d4da58.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_m21d4da58.gif

Let cos x = t 

⇒ −sinx dx = dt

When x = 0, = 1 and when

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_7c4db91c.gif

Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7796/NCERT_Solution_Math_Chapter_7_final_html_m47f8aec.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7796/NCERT_Solution_Math_Chapter_7_final_html_m7f64c05b.gif

Let 

⇒ dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7796/NCERT_Solution_Math_Chapter_7_final_html_6b4b32a9.gif

*Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7798/Item%20203_html_m6af4a309.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7798/Item%20203_html_37652a3b.gif

Let x + 1 = 

⇒ dx = dt

When x = −1, = 0 and when x = 1, = 2

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7798/Item%20203_html_5fb9abe8.gif

*Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7800/NCERT_Solution_Math_Chapter_7_final_html_6979ef1b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7800/NCERT_Solution_Math_Chapter_7_final_html_6979ef1b.gif

Let 2x = t 

⇒ 2dx = dt

When x = 1, t = 2 and when x = 2, t = 4

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7800/NCERT_Solution_Math_Chapter_7_final_html_71eddcf4.gif

*Question 9:

The value of the integral is

A. 6 B. 0 C. 3 D. 4

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7802/Item%20205_html_590694d8.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7802/Item%20205_html_61485027.gif

Let cotθ = t 

⇒ −cosec2θ dθdt

0u51v5iRCzaYuC4c78S5 pVDuyKQx0KsLPsYl5LpU5 NDLvLyKNfujZP2JR Z1YDWWsFRFeyrd5OJLH
vJzbXNj1Fx5XVHJQMKEc4v7KG6SH2yQixUivqwpKt pWuzMMNP1SxKDtI55pD0F4w

Hence, the correct answer is A.

*Question 10:

If  is

A.  cos x + x sin x

B.  x sin x

C.  x cos x

D.  sin x cos x

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7806/NCERT_Solution_Math_Chapter_7_final_html_4bad8c8f.gif

Integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7806/NCERT_Solution_Math_Chapter_7_final_html_m7c5c373f.gif

Hence, the correct answer is B.

Exercise-7.11

Question 1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8321/NCERT_Solution_Math_Chapter_7_final_html_407b64a3.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8321/NCERT_Solution_Math_Chapter_7_final_html_7f54b990.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8321/NCERT_Solution_Math_Chapter_7_final_html_319681a6.gif

Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8323/NCERT_Solution_Math_Chapter_7_final_html_54ab9afb.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8323/NCERT_Solution_Math_Chapter_7_final_html_m67aa2b1f.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8323/NCERT_Solution_Math_Chapter_7_final_html_664829e1.gif

Question 3:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8326/NCERT_Solution_Math_Chapter_7_final_html_m423eccee.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8326/NCERT_Solution_Math_Chapter_7_final_html_35986860.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8326/NCERT_Solution_Math_Chapter_7_final_html_5ec97145.gif

Question 4:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8327/NCERT_Solution_Math_Chapter_7_final_html_4a4c489.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8327/NCERT_Solution_Math_Chapter_7_final_html_32989753.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8327/NCERT_Solution_Math_Chapter_7_final_html_21644107.gif

Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8328/NCERT_Solution_Math_Chapter_7_final_html_m5de665ca.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8328/NCERT_Solution_Math_Chapter_7_final_html_5dbe20ae.gif

It can be seen that (x + 2) ≤ 0 on [−5, −2] and (x + 2) ≥ 0 on [−2, 5].

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8328/NCERT_Solution_Math_Chapter_7_final_html_773d8ee3.gif

Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8329/NCERT_Solution_Math_Chapter_7_final_html_6ad4c7e1.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8329/NCERT_Solution_Math_Chapter_7_final_html_m9a4fcf0.gif

It can be seen that (x − 5) ≤ 0on [2, 5] and (x − 5) ≥ 0 on [5, 8].

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8329/NCERT_Solution_Math_Chapter_7_final_html_m20ebff4.gif

Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8330/NCERT_Solution_Math_Chapter_7_final_html_m3484e968.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8330/NCERT_Solution_Math_Chapter_7_final_html_478c6707.gif

Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8331/NCERT_Solution_Math_Chapter_7_final_html_6c69f563.gif

Answer:

rlH7tV1rqGrQuDdTGlNGAXDDiRQOwV77jBy4qSul7GpluT9C3 by1lCIR yIp urAp3XUg HgOcYZuMSs9qTA2v4PcgktBThI1 S4jf1W5Cf5H5bE1WeJSlHyyo2VIF
EouwoS1Od2aCO4wY9uuOnPSxo3rentFaVuNMOw5l6SZJ2k3daPlAvd XUrC1wk5Izcnu toFjvZ6vGanuIMBx
VKl92hPh6Or1s7Tm7B1e3t74oMj3zJDpVUcHcyr58SI5b9ogQGnGswul3mUxtoMGLoK0QvetlT 1Yv4xN2EriUM3voPwArZLzt9EcWtQSRFJ

Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8332/NCERT_Solution_Math_Chapter_7_final_html_d58f318.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8332/NCERT_Solution_Math_Chapter_7_final_html_m26b91062.gif

Question 10:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8334/NCERT_Solution_Math_Chapter_7_final_html_m236c3cc6.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8334/NCERT_Solution_Math_Chapter_7_final_html_3eb02be4.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8334/NCERT_Solution_Math_Chapter_7_final_html_32268a59.gif

Question 11:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_66e84c6b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_mca9ceae.gif

As sin(−x) = (sin (−x))2 = (−sin x)2 = sin2x, therefore, sin2is an even function.

It is known that if f(x) is an even function, then 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_m1f03cb4.gif

Question 12:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_m4f8e3f37.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_m5ab190b8.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_m102ed4fd.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_3be073f7.gif

Question 13:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_1000a7cf.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_37730826.gif

As sin(−x) = (sin (−x))7 = (−sin x)7 = −sin7x, therefore, sin2is an odd function.

It is known that, if f(x) is an odd function, then 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_m27c8fca6.gif

Question 14:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m53f198ae.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m2ee739fb.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m6cbd1067.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m18b59200.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m1b0fe3a3.gif

Question 15:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_mb6d95c9.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_1787c899.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_m26704fd1.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_42f38cbe.gif

Question 16:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_2c1b6167.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_m17c379d8.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_2d540b86.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_6ba7be51.gif

sin (π − x) = sin x

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_71e40f52.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_5a0010e1.gif

Adding (4) and (5), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_2580c48b.gif
K0hL91SZ2bW4mGVRSwdvp1VtEPUbat U4P0mcU12eOyg JBl

Using by parts

cRLxR3XmwIFCDd GE3pxWa9zAE SfbgQIWc8zX 3vf1pQMBr9NtylZ 9Yj5qOMbPSeUi95W9mMjpNApqeRATlehJ0TTHQtY5GXilDerJeYP0SMj7D zJJ wgGD1Ma9iSYGAT fA

*Question 17:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_2ebf4e1d.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_m17123bbd.gif

It is known that, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_m4b151446.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_m67347065.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_7f16d75.gif

*Question 18:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8324/NCERT_Solution_Math_Chapter_7_final_html_m61e0a3ca.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8324/NCERT_Solution_Math_Chapter_7_final_html_m7281926a.gif

It can be seen that, (x − 1) ≤ 0 when 0 ≤ x ≤ 1 and (x − 1) ≥ 0 when 1 ≤ x ≤ 4

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8324/NCERT_Solution_Math_Chapter_7_final_html_m41aba784.gif

*Question 19:

Show that if f and g are defined as and 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_3bb51308.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_m34610f85.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_m572b38bb.gif

*Question 20:

The value of is

A. 0 B. 2 C. Π D. 1

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8333/NCERT_Solution_Math_Chapter_7_final_html_1470d63d.gif

It is known that if f(x) is an even function, then and

if f(x) is an odd function, then 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8333/NCERT_Solution_Math_Chapter_7_final_html_m125c2688.gif

Hence, the correct answer is C.

*Question 21:

The value of is

A. 2 B.  C. 0 D. uwDGD6pvZ75ZBs5 L tl0GaWcIQ5hzJ1HhOd3KLPQXs6tl0QO8beBoXTVT0d xgS3AK0wPYcMjd2G7SIlvUIr4kcO8z7U8 LKndaCNuHuQNoL6Ox

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_30dd75fb.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_m28885fd1.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_m42a37c68.gif

Hence, the correct answer is C.

Miscellaneous Exercise

*Question1:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8337/NCERT_Solution_Math_Chapter_7_final_html_m76d580f5.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8337/NCERT_Solution_Math_Chapter_7_final_html_1cd7312f.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8337/NCERT_Solution_Math_Chapter_7_final_html_m5f5406d1.gif

Equating the coefficients of x2x, and constant term, we obtain

A + B − C = 0

B + = 0

A = 1

On solving these equations, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8337/NCERT_Solution_Math_Chapter_7_final_html_m3de7c279.gif

From equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8337/NCERT_Solution_Math_Chapter_7_final_html_m578c8f06.gif

*Question 2:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7953/NCERT_Solution_Math_Chapter_7_final_html_m1df77264.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7953/NCERT_Solution_Math_Chapter_7_final_html_m4999ef5e.gif

*Question 3:

[Hint: Put]

Answer:

0kTj3Xnf1QXgr ViHFfwtK16 kk6wIsoFAxkCO87AOhGSEad9fF 6xOfyRxy5IvbFI4 iM6WTCud9vbP0fL5hVk2Zba3226yxQd0azg 4Bdtid72pktIS1q2Jl JYhDZZhTk5g
AM9DGwLsluo QGB8aVCYdRVMVSikMpvBKuB6mSBVf7RKlvQA1A7EnGA7SvgGBE3nKO9SUgVfAVX1gWFPDeAd Hqk1 GkHCZNgZaABRaI3VwQKgKiMg2YDHo0Xu0LzkuP6pmOIwg

*Question 4:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7948/NCERT_Solution_Math_Chapter_7_final_html_5dacd89f.gif

Answer:

oF 1an66lj2nbv 1pBTsOaHud8fiN4HaGIkQS gtk24M1k5j8lN9niczU2k3F K6H3m YcPWP8vu 05Z8D0NuqWG TcvACRUH6OEI1hhnYuJtFfXh8B3UiW7lK2FVT8UT4mm M
omCK1gzAjW4Q9smqD9qE5j2NBRcCzZ3 59DSIZqNA LoErRLOzkXb96ZZ1SgiZwnftJDo72cQrik U0T0A TMnqqITFsglNg0Mzve9JgHYPmIqF0 W8ylVsFjVg50VaNjV4j8cg

*Question 5:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7949/NCERT_Solution_Math_Chapter_7_final_html_m1c750f15.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7949/NCERT_Solution_Math_Chapter_7_final_html_4257a166.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7949/NCERT_Solution_Math_Chapter_7_final_html_m61d78b39.gif

On dividing, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7949/NCERT_Solution_Math_Chapter_7_final_html_2b0e959c.gif

*Question 6:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7950/NCERT_Solution_Math_Chapter_7_final_html_77c6b083.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7950/NCERT_Solution_Math_Chapter_7_final_html_m1d1e21a2.gif

Equating the coefficients of x2x, and constant term, we obtain

A + B = 0

C = 5

9A + = 0

On solving these equations, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7950/NCERT_Solution_Math_Chapter_7_final_html_349422a5.gif

From equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7950/NCERT_Solution_Math_Chapter_7_final_html_578f8b8e.gif

*Question 7:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7951/NCERT_Solution_Math_Chapter_7_final_html_m1249c9d6.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7951/NCERT_Solution_Math_Chapter_7_final_html_m1249c9d6.gif

Let  a t

⇒ dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7951/NCERT_Solution_Math_Chapter_7_final_html_2b521cac.gif

*Question 8:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7952/NCERT_Solution_Math_Chapter_7_final_html_40aaf85b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7952/NCERT_Solution_Math_Chapter_7_final_html_471dd3f.gif

*Question 9:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7918/NCERT_Solution_Math_Chapter_7_final_html_m3fc14e9.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7918/NCERT_Solution_Math_Chapter_7_final_html_m3fc14e9.gif

Let sin x = t 

⇒ cos x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7918/NCERT_Solution_Math_Chapter_7_final_html_m6cb4b391.gif

*Question 10:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7922/NCERT_Solution_Math_Chapter_7_final_html_m483003f0.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7922/NCERT_Solution_Math_Chapter_7_final_html_474a8251.gif

*Question 11:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7924/NCERT_Solution_Math_Chapter_7_final_html_m52b54278.gif

Answer:

QPm
k2iURQwdCLSJS 1qut dSwWRV 4qEOJbcb1CdEmrmrxeOl cecNL39stvwzQkwhKgGFsXIWevgD4pRcqJOG1XqipodN0nZ3uxC7rxfmu45bf3fdXinfM43DUD VKLEDSK6kZy0k

*Question 12:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7928/NCERT_Solution_Math_Chapter_7_final_html_m735afaad.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7928/NCERT_Solution_Math_Chapter_7_final_html_m735afaad.gif

Let x= t 

⇒ 4x3 dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7928/NCERT_Solution_Math_Chapter_7_final_html_m6d3e2293.gif

*Question 13:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7930/NCERT_Solution_Math_Chapter_7_final_html_2df2b31c.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7930/NCERT_Solution_Math_Chapter_7_final_html_2df2b31c.gif

Let ex = t 

⇒ ex dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7930/NCERT_Solution_Math_Chapter_7_final_html_36b2aedd.gif

*Question 14:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7954/NCERT_Solution_Math_Chapter_7_final_html_1cd0595e.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7954/NCERT_Solution_Math_Chapter_7_final_html_1e1a0b82.gif

Equating the coefficients of x3x2x, and constant term, we obtain

A + C = 0

B + D = 0

4A + C = 0

4D = 1

On solving these equations, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7954/NCERT_Solution_Math_Chapter_7_final_html_m35f98eec.gif

From equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7954/NCERT_Solution_Math_Chapter_7_final_html_m3c6c133d.gif

*Question 15:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7955/NCERT_Solution_Math_Chapter_7_final_html_m4a699e43.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7955/NCERT_Solution_Math_Chapter_7_final_html_m4a699e43.gif= cos3 x × sin x

Let cos x = t 

⇒ −sin x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7955/NCERT_Solution_Math_Chapter_7_final_html_m2f46a304.gif

*Question 16:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7956/NCERT_Solution_Math_Chapter_7_final_html_m62ecd5df.gif

Answer:

By9JxKatjtJJ4GokX0mE4AnQHm3ikViU3BSemflNaHUQgmwWpxfiy1 44YOYW8rB7yd9Vpc7YT067HwCoIr000 eNAJSoONHdAaTYbFV 7Xfs BywRaQXwX5AmS2R1KCifEqzIk

*Question 17:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7957/NCERT_Solution_Math_Chapter_7_final_html_m12636567.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7957/NCERT_Solution_Math_Chapter_7_final_html_1c9df5fd.gif

*Question 18:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7958/NCERT_Solution_Math_Chapter_7_final_html_fc4380a.gif

Answer:

NnygJnVOyIK8CFVdOjtGW8AoTdmSlQTIQEin5YuWGQACi2wZC8Y0lYCKcrsBWmiqXOaEutfV0XxtlP4W4 v Ngz 9wmPXYuUht3NMU9uEtm4u2mo7yN GfaD0zbQJZMJ5vrigwg
P2q8cTvcRZaaubTd kN1peaYu1LoM6BdHW kcD3II F7JmSLo9FjHpjkqjO4 VZsOM nehm6rGG4tup30hzfcjPeGSncQsDV0QC4SaTX7wMnq1VC5vt q9zuh3Y6hto ywzYs

*Question 19:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7959/NCERT_Solution_Math_Chapter_7_final_html_m633ccb7.gif

Answer:

UsQP2GCZUlIjJEa3OjL4ZUyQ4hdcPNDQOlT9Uz hTefV2KuhMnXUYeQKnzCILucf8N5MgNHPyO9eLqux8ZgeH6N8HM5CWKsr4JKuZdVl87GO4Ks5yGBzTREjxoEOZQab97 aCiQ
gxaWL89E9bLQ sargZbR4p2hpk8TNR1
vrSRSde8nmxaIfBtyjwIR d8VrIV C6e yGPG MGvv5 OjtHAzrhlUK1yezU F svviLTHWcn7LkJUJMBXKo7AQwHrGjMcHkDRg8JjAUQlPwKPnOViuy1p2vs0 abzovl2N2vlY

….(i)

= taking

Let

5ofYJ5as3g60Bto6EVuPBTSimugF5wy6xQByt8WEUbV9zotr d7T51LFw8XU C8 6IVvb1aEdtNXHlSrJNZwMeKwbjEsbDpvW3SAle 8CG9RFt7UcRMNzG3U0e4HP9lo519p7tA

∴ Integral

rJnO ucxooNAu gbfgKTyy1wYMNrCpZD2Tad5ATnhPvMuIskMb66ygAAGejGlhiQVKZWqHte0blZ7lgAYPoDEbbSBVxYS7R
OoSAFWlzDsKxP9UdzQJwcxGN1R2cZ6gpVpbSgzt Bmp5qqdMHNt0mC8qddF4MLiQ 5zMHSXFL3PS0Rxi7m4bOxy2 guHvMidgds2Ie2N2pnLskXG2c4ZO2EAAs8B0PFXLmh21Yw
PJsLSJFc3PwD2a3qlWMMctHkW8RJbhWHm9YBPtaGOKrFY9PjlZjZIQVDcd RlX4zSjGgkt56zduy9OVdCC3OoiG1Gow5KPietpeYj0zzBo6I wl98sQdvYGQ0zwOV0bC8v93i40

….(2)

From (1) and (2)

OutuEKktHnKVP7pOrZlHxmKFluhcaVJdH5UAgXG96R Ky9iOuTR 78Q1RyPo27N76d9GHPqObO4 o
NatGlfcuxS8wEb4soaCQ09areF1JIqEqsrTbTur1KNX ZGCX4JpPVnsgKwJbfmvPESr rSsrU6of4UlW30MdbmlRTwOHZ9cB aNbHUtsUrWPzj7ljnMp8xy vFLSx OlwkP9ol0

*Question 20:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7960/NCERT_Solution_Math_Chapter_7_final_html_50604b7.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7960/NCERT_Solution_Math_Chapter_7_final_html_762a0d1a.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7960/NCERT_Solution_Math_Chapter_7_final_html_3e40ed34.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7960/NCERT_Solution_Math_Chapter_7_final_html_m1c3fbf3.gif

*Question 21:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7961/NCERT_Solution_Math_Chapter_7_final_html_m4b469c28.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7961/NCERT_Solution_Math_Chapter_7_final_html_2b1a77ef.gif

*Question 22:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7962/NCERT_Solution_Math_Chapter_7_final_html_69559a26.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7962/NCERT_Solution_Math_Chapter_7_final_html_62dfc439.gif

Equating the coefficients of x2x,and constant term, we obtain

A + C = 1

3A + B + 2= 1

2A + 2B + C = 1

On solving these equations, we obtain

A = −2, B = 1, and C = 3

From equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7962/NCERT_Solution_Math_Chapter_7_final_html_m6aa0e950.gif

*Question 23:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7963/NCERT_Solution_Math_Chapter_7_final_html_m5796614.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7963/NCERT_Solution_Math_Chapter_7_final_html_m2e79a1c8.gif

*Question 24:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7964/NCERT_Solution_Math_Chapter_7_final_html_m7bddd4ea.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7964/NCERT_Solution_Math_Chapter_7_final_html_m56d31346.gif

Integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7964/NCERT_Solution_Math_Chapter_7_final_html_m636f9379.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7964/NCERT_Solution_Math_Chapter_7_final_html_49b75c6e.gif

*Question 25:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7965/NCERT_Solution_Math_Chapter_7_final_html_m54750f4f.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7965/NCERT_Solution_Math_Chapter_7_final_html_37560177.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7965/NCERT_Solution_Math_Chapter_7_final_html_33bd321a.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7965/NCERT_Solution_Math_Chapter_7_final_html_142ee072.gif

*Question 26:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7966/Items%20253_html_3a7081af.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7966/Items%20253_html_m59f605f1.gif

When = 0, = 0 and  when 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7966/Items%20253_html_4557600.gif

*Question 27:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7967/NCERT_Solution_Math_Chapter_7_final_html_m4683c3a.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7967/NCERT_Solution_Math_Chapter_7_final_html_4d9d851e.gif

When and when

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7967/NCERT_Solution_Math_Chapter_7_final_html_4e934ca9.gif

*Question 28:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7968/NCERT_Solution_Math_Chapter_7_final_html_m4441a09b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7968/NCERT_Solution_Math_Chapter_7_final_html_m7484db7e.gif

When and when 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7968/NCERT_Solution_Math_Chapter_7_final_html_480ad49.gif

As  therefore, is an even function.

It is known that if f(x) is an even function, then 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7968/NCERT_Solution_Math_Chapter_7_final_html_m48c28a40.gif

*Question 29:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7969/NCERT_Solution_Math_Chapter_7_final_html_m776c0f8a.gif

Answer:

8TNkfU6nTt euAEV4HBYU9qi4h0GnZuHcuLJgBoFwUSEno hU4Y1idEj2Z

*Question 30:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7970/NCERT_Solution_Math_Chapter_7_final_html_m52d730c.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7970/NCERT_Solution_Math_Chapter_7_final_html_177cac27.gif

*Question 31:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7971/Items%20258_html_m33842e5c.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7971/Items%20258_html_m75708a03.gif

From equation (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7971/Items%20258_html_65374166.gif

*Question 32:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7972/Items%20259_html_m2f373ec4.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7972/Items%20259_html_30ba894a.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7972/Items%20259_html_m4e0d6c68.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7972/Items%20259_html_328fcc21.gif

*Question 33:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_mae4faef.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_21bf0800.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_m1ab518e9.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_m2dc72d4.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_37eba167.gif

From equations (1), (2), (3), and (4), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_m5ef1a4ac.gif

*Question 34:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7974/NCERT_Solution_Math_Chapter_7_final_html_m48898861.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7974/NCERT_Solution_Math_Chapter_7_final_html_219c8b36.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7974/NCERT_Solution_Math_Chapter_7_final_html_6c9a2099.gif

Equating the coefficients of x2x, and constant term, we obtain

A + C = 0

A + B = 0

B = 1

On solving these equations, we obtain

A = −1, C = 1, and B = 1

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7974/NCERT_Solution_Math_Chapter_7_final_html_4c7a449f.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7974/NCERT_Solution_Math_Chapter_7_final_html_m336ec5df.gif

Hence, the given result is proved.

*Question 35:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7975/NCERT_Solution_Math_Chapter_7_final_html_1dd3b058.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7975/NCERT_Solution_Math_Chapter_7_final_html_m202c3c87.gif

Integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7975/NCERT_Solution_Math_Chapter_7_final_html_6e440035.gif

Hence, the given result is proved.

*Question 36:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7977/NCERT_Solution_Math_Chapter_7_final_html_7f84818e.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7977/NCERT_Solution_Math_Chapter_7_final_html_678e1db4.gif

Therefore, f (x) is an odd function.

It is known that if f(x) is an odd function, then 

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7977/NCERT_Solution_Math_Chapter_7_final_html_m7faee91f.gif

Hence, the given result is proved.

*Question 37:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7978/NCERT_Solution_Math_Chapter_7_final_html_41b38be1.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7978/NCERT_Solution_Math_Chapter_7_final_html_148e804.gif

Hence, the given result is proved.

*Question 38:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7979/NCERT_Solution_Math_Chapter_7_final_html_m53d9d2bb.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7979/NCERT_Solution_Math_Chapter_7_final_html_m1711030b.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7979/NCERT_Solution_Math_Chapter_7_final_html_3fb92cfd.gif

Hence, the given result is proved.

*Question 39:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7980/NCERT_Solution_Math_Chapter_7_final_html_54483698.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7980/NCERT_Solution_Math_Chapter_7_final_html_m23aef4eb.gif

Integrating by parts, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7980/NCERT_Solution_Math_Chapter_7_final_html_7ba2d404.gif

Let 1 − x2 = t 

⇒ −2x dx = dt

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7980/NCERT_Solution_Math_Chapter_7_final_html_m264f525f.gif

Hence, the given result is proved.

*Question 40:

Evaluate as a limit of a sum.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_m1e3a7643.gif

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_4cd0ad5d.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_1f641d48.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_724206cf.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_743088e5.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_42c5fb6f.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_m1d159455.gif

*Question 41:

is equal to

A.  B. 

C.  D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7982/NCERT_Solution_Math_Chapter_7_final_html_m6eb6c002.gif

Hence, the correct answer is A.

*Question 42:

is equal to

A.  B. 

C.  D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7943/NCERT_Solution_Math_Chapter_7_final_html_m28e59cdb.gif

Hence, the correct answer is B.

*Question 43:

If then is equal to

A.  B. 

C.  D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7939/NCERT_Solution_Math_Chapter_7_final_html_m6cc05e7b.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7939/NCERT_Solution_Math_Chapter_7_final_html_m1930c56c.gif

Hence, the correct answer is D.

*Question 44:

The value of is

A. 1 B. 0 C. ZJ03UeyJzdFtkRMFrp0LMGlTFyDUJ6z2H7rMUN9ZF5pysXQRmhdY5g6jjtcXm5waxoHsQ7yxUfPsWApH1u 0xKWLEA D. 

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7935/NCERT_Solution_Math_Chapter_7_final_html_m470456ac.gif
https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7935/NCERT_Solution_Math_Chapter_7_final_html_1361c568.gif

Adding (1) and (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7935/NCERT_Solution_Math_Chapter_7_final_html_m7743de6a.gif

Hence, the correct answer is B.

Frequently Asked Questions on NCERT Solutions for Class 12 Maths Chapter 7

What is the meaning of integrals in NCERT Solutions for Class 12 Maths Chapter 7?

The area under a curve is often referred to as integral in Calculus. It can be represented on a graph, as a mathematical function. List of important formulas for determining the integration of various functions is covered in this chapter. Integral is also known as antiderivative and we also know that integration formulas of some functions are the reverse of their differentiation formulas. This chapter is also considered to be one of the important chapters from the exam point of view.

Is Chapter 7 of NCERT Solutions for Class 12 Maths difficult to understand?

No, Chapter 7 of NCERT Solutions for Class 12 Maths is not difficult to understand. It is one of the interesting topics in Class 12 which would be continued in higher levels of education also. Good knowledge of integral formulas will make it easier for the students to solve the basic sums of integration efficiently. A proper knowledge of derivatives is necessary to understand the concepts of integral calculus without any difficulty.

Where can I download the NCERT Solutions for Class 12 Maths Chapter 7 PDF online?

You can find the NCERT Solutions for Class 12 Maths Chapter 7 on SWC. The solutions are prepared by the highly experienced faculty having vast conceptual knowledge. These are considered to be one of the best rated solutions available online. All the problems from the NCERT textbook are solved in a step wise manner based on the latest syllabus and exam pattern designed by the CBSE board.

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NCERT Solutions Class 12 Maths Chapters

  • Chapter 1 Relations and Function
  • Chapter 2 Inverse Trigonomtry
  • Chapter 3 Matrices
  • Chapter 4 Determinants
  • Chapter 5 Continuity and Differentiability
  • Chapter 6 Applications of Derivatives
  • Chapter 7 Integrals
  • Chapter 8 Application of Integrals
  • Chapter 9 Differential Equations
  • Chapter 10 Vector
  • Chapter 11 Three Dimensional Geometry
  • Chapter 12 Linear Programming
  • Chapter 13 Probability

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