NCERT Solution for Class 12 Mathematics Chapter 7, “Integrals,” is an essential study material designed to help students understand the fundamental concepts of integral calculus. Swastik Classes, a leading coaching institute, has developed comprehensive NCERT solutions that provide step-by-step explanations and solved examples to help students develop a deeper understanding of the subject. The chapter covers topics such as properties of integrals, methods of integration, integration using partial fractions, and integration using trigonometric identities. With the help of Swastik Classes’ NCERT solutions, students can improve their problem-solving skills and gain the confidence to tackle complex integral problems. These solutions are also useful for students who are preparing for competitive exams like JEE, NEET, and other entrance exams. Overall, Swastik Classes’ NCERT Solution for Class 12 Mathematics Chapter 7 is an essential resource for students who want to excel in mathematics and build a strong foundation in integral calculus.

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**Answers of Mathematics NCERT solutions for class 12 Chapter 7 Integrals**

**Chapter 7**

**Integrals**

**Exercise-7.1**

**Question 1:**

sin 2*x*

**Answer:**

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

It is known that,

Therefore, the anti-derivative of is

**Question 2:**

Cos 3*x*

**Answer:**

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

It is known that,

Therefore, the anti-derivative of is .

**Question 3:**

*e*^{2}*x*

**Answer:**

The anti-derivative of *e*^{2}*x *is the function of* x* whose derivative is *e*^{2}*x*.

It is known that,

Therefore, the anti-derivative of is

**Question 4:**

**Answer:**

The anti-derivative of

is the function of *x *whose derivative is

It is known that,

Therefore, the anti-derivative of is

**Question 5:**

**Answer:**

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

It is known that,

Therefore, the anti-derivative of is .

**Question 6:**

**Answer:**

**Question 7:**

**Answer:**

**Question 8:**

**Answer:**

**Question 9:**

**Answer:**

**Question 10:**

**Answer:**

**Question 11:**

**Answer:**

**Question 12:**

**Answer:**

**Question 13:**

**Answer:**

On dividing, we obtain

**Question 14:**

**Answer:**

**Question 15:**

**Answer:**

**Question 16:**

**Answer:**

**Question 17:**

**Answer:**

**Question 18:**

**Answer:**

**Question 19:**

**Answer:**

**Question 20:**

**Answer:**

**Question 21:**

The anti-derivative of equals

**(A)** **(B) **

**(C) ** **(D) **

**Answer:**

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,

∴Anti derivative of

Also,

Hence, the correct answer is A.

**Exercise-7.2**

**Question 1:**

**Answer:**

Let = *t*

∴2*x dx* = *dt*

**Question 2:**

**Answer:**

Let log |*x*| = *t*

∴

**Question 3:**

**Answer:**

Let 1 + log *x *= *t*

∴

**Question 4:**

sin *x* ⋅ sin (cos *x*)

**Answer:**

sin *x* ⋅ sin (cos *x*)

Let cos *x* = *t*

∴ −sin *x dx = dt*

**Question 5:**

**Answer:**

Let

∴ 2*adx = dt*

**Question 6:**

**Answer:**

Let *ax + b = t*

⇒ *adx = dt*

**Question 7:**

**Answer:**

Let

∴ *dx = dt*

**Question 8:**

**Answer:**

Let 1 + 2*x*^{2} = *t*

∴ 4*xdx = dt*

**Question 9:**

**Answer:**

Let

∴ (2*x* + 1)*dx = dt*

**Question 10:**

**Answer:**

Let

∴

**Question 11:**

**Answer:**

Put

**Question 12:**

**Answer:**

Let

∴

**Question 13:**

**Answer:**

Let

∴ 9*x*^{2} *dx = dt*

**Question 14:**

**Answer:**

Let log *x = t*

∴

**Question 15:**

**Answer:**

Let

∴ −8*x dx* = *dt*

**Question 16:**

**Answer:**

Let

∴ 2*dx = dt*

**Question 17:**

**Answer:**

Let

∴ 2*xdx = dt*

**Question 18:**

**Answer:**

Let

∴

**Question 19:**

**Answer:**

Dividing numerator and denominator by *ex*, we obtain

Let

∴

**Question 20:**

**Answer:**

Let

∴

**Question 21:**

**Answer:**

Put

**Question 22:**

**Answer:**

Let 7 − 4*x* = *t*

∴ −4*dx = dt*

**Question 23:**

**Answer:**

Let

∴

**Question 24:**

**Answer:**

Let

∴

**Question 25:**

**Answer:**

Let

∴

**Question 26:**

**Answer:**

Let

∴

**Question 27:**

**Answer:**

Let sin 2*x* = *t*

∴

**Question 28:**

**Answer:**

Let

∴ cos *x dx* = *dt*

**Question 29:**

cot *x* log sin *x*

**Answer:**

Let log sin *x* = *t*

**Question 30:**

**Answer:**

Let 1 + cos *x = t*

∴ −sin *x dx* = *dt*

**Question 31:**

**Answer:**

Let 1 + cos *x* = *t*

∴ −sin *x* *dx = dt*

**Question 32:**

**Answer:**

Let sin *x* + cos *x* = *t*

⇒ (cos *x* − sin *x*) *dx* = *dt*

**Question 33:**

**Answer:**

Put cos *x* − sin *x* = *t*

⇒ (−sin *x* − cos *x*) *dx* = *dt*

***Question 34:**

**Answer:**

***Question 35:**

**Answer:**

Let 1 + log *x* = *t*

∴

***Question 36:**

**Answer:**

Let** **

∴

***Question 37:**

**Answer:**

Let *x*^{4} = *t*

∴ 4*x*^{3}* dx = dt*

Let

∴

From (1), we obtain

**Question 38:**

equals

**Answer:**

Let

∴

Hence, the correct answer is D.

**Question 39:**

equals

**A.** **B.**

**C.** **D. **

**Answer:**

Hence, the correct answer is B.

**Exercise-7.3**

**Question 1:**

**Answer:**

**Question 2:**

**Answer:**

It is known that,

**Question 3:**

cos 2*x* cos 4*x* cos 6*x*

**Answer:**

It is known that,

**Question 4:**

sin^{3} (2*x* + 1)

**Answer:**

Let

**Question 5:**

sin^{3} *x* cos^{3} *x*

**Answer:**

**Question 6:**

sin *x* sin 2*x* sin 3*x*

**Answer:**

It is known that,

**Question 7:**

sin 4*x* sin 8*x*

**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:**

**Answer:**

**Question 9:**

**Answer:**

**Question 10:**

sin^{4} *x*

**Answer:**

**Question 11:**

cos^{4} 2*x*

**Answer:**

**Question 12:**

**Answer:**

**Question 13:**

**Answer:**

**Question 14:**

**Answer:**

**Question 15:**

**Answer:**

**Question 16:**

tan^{4}*x*

**Answer:**

From equation (1), we obtain

**Question 17:**

**Answer:**

**Question 18:**

**Answer:**

**Question 19:**

**Answer:**

***Question 20:**

**Answer:**

***Question 21:**

sin^{−1} (cos* x*)

**Answer:**

It is known that,

Substituting in equation (1), we obtain

***Question 22:**

**Answer:**

***Question 23:**

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:**

Hence, the correct answer is A.

***Question 24:**

equals

**A.**

**B.**

**C.**

**D. **

**Answer:**

Let (*e^x).x* = *t*

Hence, the correct answer is B.

**Exercise-7.4**

**Question 1:**

**Answer:**

Let *x*^{3} = *t*

∴ 3*x*^{2} *dx* = *dt*

**Question 2:**

**Answer:**

Let 2*x* = *t*

∴ 2*dx* = *dt*

**Question 3:**

**Answer:**

Let 2 − *x *= *t*

⇒ −*dx* = *dt*

**Question 4:**

**Answer:**

Let 5*x* =* t*

∴ 5*dx* = *dt*

**Question 5:**

**Answer:**

**Question 6:**

**Answer:**

Let *x*^{3} = *t*

∴ 3*x*^{2} *dx* = *dt*

**Question 7:**

**Answer:**

From (1), we obtain

**Question 8:**

**Answer:**

Let *x*^{3} = *t*

⇒ 3*x*^{2} *dx* = *dt*

**Question 9:**

**Answer:**

Let tan *x* =* t*

∴ sec^{2}*x* *dx* = *dt*

**Question 10:**

**Answer:**

**Question 11:**

**Answer:**

=

Let (3x+1)=t

∴3 dx = dt

⇒

=(1/3)

**Question 12:**

**Answer:**

**Question 13:**

**Answer:**

**Question 14:**

**Answer:**

**Question 15:**

**Answer:**

**Question 16:**

**Answer:**

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

4*A* = 4 ⇒ *A* = 1

*A* + *B* = 1 ⇒ *B* = 0

Let 2*x*^{2} + *x* − 3 = *t*

∴ (4*x* + 1) *dx *= *dt*

**Question 17:**

**Answer:**

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

From (1), we obtain

From equation (2), we obtain

**Question 18:**

**Answer:**

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

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

**Question 19:**

**Answer:**

Equating the coefficients of *x* and constant term, we obtain

2*A* = 6 ⇒ *A* = 3

−9*A* + *B* = 7 ⇒ *B* = 34

∴ 6*x* + 7 = 3 (2*x* − 9) + 34

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

**Question 20:**

**Answer:**

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

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

**Question 21:**

**Answer:**

Let *x*^{2} + 2*x* +3 = *t*

⇒ (2*x* + 2) *dx* =*dt*

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

***Question 22:**

**Answer:**

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

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

***Question 23:**

**Answer:**

Equating the coefficients of *x* and constant term, we obtain

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

***Question 24:**

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:**

Hence, the correct answer is B.

***Question 25:**

equals

**A.**

**B.**

**C.**

**D. **

**Answer:**

Hence, the correct answer is B.

**Exercise-7.5**

**Question 1:**

**Answer:**

Let

Equating the coefficients of *x* and constant term, we obtain

*A* + *B *= 1

2*A* +* B *= 0

On solving, we obtain

*A* = −1 and *B* = 2

**Question 2:**

**Answer:**

Let

Equating the coefficients of *x* and constant term, we obtain

*A* +* B *= 0

−3*A* + 3*B* = 1

On solving, we obtain

**Question 3:**

**Answer:**

Let

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

*A* = 1, *B* = −5, and *C* = 4

**Question 4:**

**Answer:**

Let

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

**Question 5:**

**Answer:**

Let

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

*A* = −2 and *B* = 4

**Question 6:**

**Answer:**

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

Therefore, on dividing (1 − *x*^{2}) by *x*(1 − 2*x*), we obtain

Let

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

*A *= 2 and* B *= 3

Substituting in equation (1), we obtain

**Question 7:**

**Answer:**

Let

Equating the coefficients of *x*^{2}, *x*, and constant term, we obtain

*A* + *C* = 0

−*A* + *B* = 1

−*B* + *C* = 0

On solving these equations, we obtain

From equation (1), we obtain

**Question 8:**

**Answer:**

Let

Substituting *x* = 1, we obtain

Equating the coefficients of *x*^{2} and constant term, we obtain

*A* + *C* = 0

−2*A* + 2*B* + *C* = 0

On solving, we obtain

**Question 9:**

**Answer:**

Let

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

*B* = 4

Equating the coefficients of *x*^{2} and *x*, we obtain

*A* + *C* = 0

*B* − 2*C* = 3

On solving, we obtain

**Question 10:**

**Answer:**

Let

Equating the coefficients of *x*^{2} and *x*, we obtain

**Question 11:**

**Answer:**

Let

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

**Question 12:**

**Answer:**

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

Therefore, on dividing (*x*^{3} +* x *+ 1) by *x*^{2} − 1, we obtain

Let

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

**Question 13:**

**Answer:**

Equating the coefficient of *x*^{2}, *x*, 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

**Question 14:**

**Answer:**

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

*A* = 3

2*A* + *B *= −1 ⇒ *B* = −7

**Question 15:**

**Answer:**

Equating the coefficient of *x*^{3}, *x*^{2},* x*, and constant term, we obtain

On solving these equations, we obtain

**Question 16:**

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

**Answer:**

Multiplying numerator and denominator by , we obtain

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

*A* = 1 and *B* = −1

**Question 17:**

[Hint: Put sin *x* = *t*]

**Answer:**

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

*A* = 1 and *B* = −1

**Question 18:**

**Answer:**

Equating the coefficients of *x*^{3}, *x*^{2}, *x*, and constant term, we obtain

*A* + *C* = 0

*B* + *D* = 4

4*A* + 3*C* = 0

4*B* + 3*D* = 10

On solving these equations, we obtain

*A* = 0, *B* = −2, *C* = 0, and *D* = 6

***Question 19:**

**Answer:**

Let *x*^{2} = *t*

⇒ 2*x* *dx* = *dt*

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

***Question 20:**

**Answer:**

Multiplying numerator and denominator by *x*^{3}, we obtain

Let *x*^{4} =* t*

⇒ 4*x*^{3}*dx* = *dt*

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

*A* = −1 and *B* = 1

***Question 21:**

[Hint: Put *ex* = *t*]

**Answer:**

Let *ex* = *t *

⇒ *ex* *dx* = *dt*

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

*A* = −1 and *B* = 1

***Question 22:**

**A.** **B.**

**C.** **D.**

**Answer:**

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

*A* = −1 and *B* = 2

Hence, the correct answer is B.

***Question 23:**

**A. ** **B. **

**C. ** **D.**

**Answer:**

Equating the coefficients of *x*^{2}, *x*, and constant term, we obtain

*A* + *B* = 0

*C* = 0

*A* = 1

On solving these equations, we obtain

*A *= 1, *B* = −1, and *C* = 0

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

**Question 2:**

**Answer:**

Let

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

**Question 3:**

**Answer:**

Let

Taking *x*^{2} as first function and *ex* as second function and integrating by parts, we obtain

Again, integrating by parts, we obtain

**Question 4:**

*x* log*x*

**Answer:**

Let

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

**Question 5:**

*x* log 2*x*

**Answer:**

Let

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

**Question 6:**

*x*^{2 }log *x*

**Answer:**

Let

Taking log *x* as first function and *x*^{2} as second function and integrating by parts, we obtain

**Question 7:**

**Answer:**

Let

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

**Question 8:**

**Answer:**

Let

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

**Question 9:**

**Answer:**

Let

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

**Question 10:**

**Answer:**

Let

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

**Question 11:**

**Answer:**

Let

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

**Question 12:**

**Answer:**

Let

Taking *x* as first function and sec^{2}*x* as second function and integrating by parts, we obtain

**Question 13:**

**Answer:**

Let

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

**Question 14:**

**Answer:**

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

Again, integrating by parts, we obtain

**Question 15:**

**Answer:**

Let

Let *I* = *I*_{1} + *I*_{2} … (1)

Where,

and

Taking log *x* as first function and *x*^{2 }as second function and integrating by parts, we obtain

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

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

**Question 16:**

**Answer:**

Let

Let

⇒

∴

It is known that

,

**Question 17:**

**Answer:**

Let

Let

⇒

It is known that,

**Question 18:**

**Answer:**

Let

It is known that,

From equation (1), we obtain

**Question 19:**

**Answer:**

Also, let

It is known that,

***Question 20:**

**Answer:**

Let

⇒

It is known that,

***Question 21:**

**Answer:**

Let

Integrating by parts, we obtain

Again, integrating by parts, we obtain

***Question 22:**

**Answer:**

Let

= 2*θ*

Integrating by parts, we obtain

***Question 23:**

equals

**Answer:**

Let

Also, let

⇒

Hence, the correct answer is A.

***Question 24:**

equals

**Answer:**

Let

Also, let

⇒

It is known that,

Hence, the correct answer is B.

**Exercise-7.7**

**Question 1:**

**Answer:**

**Question 2:**

**Answer:**

**Question 3:**

**Answer:**

**Question 4:**

**Answer:**

**Question 5:**

**Answer:**

**Question 6:**

**Answer:**

**Question 7:**

**Answer:**

***Question 8:**

**Answer:**

***Question 9:**

**Answer:**

***Question 10:**

is equal to

**A.** **B.**

**C.** **D. **

**Answer:**

Hence, the correct answer is A.

***Question 11:**

is equal to

**A.**

**B.**

**C.**

**D. **

**Answer:**

Hence, the correct answer is D.

**Exercise-7.8**

**Question 1:**

**Answer:**

It is known that,

**Question 2:**

**Answer:**

It is known that,

**Question 3:**

**Answer:**

It is known that,

**Question 4:**

**Answer:**

It is known that,

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

**Question 5:**

**Answer:**

It is known that,

**Question 6:**

**Answer:**

It is known that,

**Exercise-7.9**

**Question 1:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 2:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 3:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 4:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 5:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 6:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 7:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 8:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 9:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 10:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 11:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 12:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 13:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 14:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 15:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 16:**

**Answer:**

Let

Equating the coefficients of *x* and constant term, we obtain

A = 10 and B = −25

Substituting the value of *I*_{1} in (1), we obtain

**Question 17:**

**Answer:**

By second fundamental theorem of calculus, we obtain

**Question 18:**

**Answer:**

By second fundamental theorem of calculus, we obtain

***Question 19:**

**Answer:**

By second fundamental theorem of calculus, we obtain

***Question 20:**

**Answer:**

By second fundamental theorem of calculus, we obtain

***Question 21:**

equals

**A.** **B.** **C.** **D. **

**Answer:**

By second fundamental theorem of calculus, we obtain

Hence, the correct answer is D.

***Question 22:**

equals

**A.** **B.** **C.** **D. **

**Answer:**

By second fundamental theorem of calculus, we obtain

Hence, the correct answer is C.

**Exercise-7.10**

**Question 1:**

**Answer:**

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

**Question 2:**

**Answer:**

Also, let

**Question 3:**

**Answer:**

Also, let *x* = tan*θ*

⇒ *dx* = sec^{2}*θ* d*θ*

When *x* = 0, *θ* = 0 and when *x *= 1,

Takingas first function and sec^{2}*θ* as second function and integrating by parts, we obtain

**Question 4:**

**Answer:**

Let *x *+ 2 = *t*^{2}

⇒ *dx *= 2*tdt*

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

**Question 5:**

**Answer:**

Let cos *x* = *t*

⇒ −sin*x* *dx* = *dt*

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

**Question 6:**

**Answer:**

Let

⇒ *dx* = *dt*

***Question 7:**

**Answer:**

Let *x* + 1 = *t *

⇒ *dx* = *dt*

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

***Question 8:**

**Answer:**

Let 2*x* =* t*

⇒ 2*dx* = *dt*

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

***Question 9:**

The value of the integral is

**A. **6 **B. **0 **C. **3 **D.** 4

**Answer:**

Let cot*θ* =* t *

⇒ −cosec2*θ dθ*= *dt*

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 *+ *x* cos *x*

**Answer:**

Integrating by parts, we obtain

Hence, the correct answer is B.

**Exercise-7.11**

**Question 1:**

**Answer:**

Adding (1) and (2), we obtain

**Question 2:**

**Answer:**

Adding (1) and (2), we obtain

**Question 3:**

**Answer:**

Adding (1) and (2), we obtain

**Question 4:**

**Answer:**

Adding (1) and (2), we obtain

**Question 5:**

**Answer:**

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

**Question 6:**

**Answer:**

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

**Question 7:**

**Answer:**

**Question 8:**

**Answer:**

**Question 9:**

**Answer:**

**Question 10:**

**Answer:**

Adding (1) and (2), we obtain

**Question 11:**

**Answer:**

As sin^{2 }(−*x*) = (sin (−*x*))^{2} = (−sin *x*)^{2} = sin^{2}*x*, therefore, sin^{2}*x *is an even function.

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

**Question 12:**

**Answer:**

Adding (1) and (2), we obtain

**Question 13:**

**Answer:**

As sin^{7 }(−*x*) = (sin (−*x*))^{7} = (−sin *x*)^{7} = −sin^{7}*x*, therefore, sin^{2}*x *is an odd function.

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

**Question 14:**

**Answer:**

It is known that,

**Question 15:**

**Answer:**

Adding (1) and (2), we obtain

**Question 16:**

**Answer:**

Adding (1) and (2), we obtain

sin (π − *x*) = sin *x*

Adding (4) and (5), we obtain

Using by parts

***Question 17:**

**Answer:**

It is known that,

Adding (1) and (2), we obtain

***Question 18:**

**Answer:**

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

***Question 19:**

Show that if *f* and *g* are defined as and

**Answer:**

Adding (1) and (2), we obtain

***Question 20:**

The value of is

**A. **0 **B. **2 **C. **Π **D.** 1

**Answer:**

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

if *f*(*x*) is an odd function, then

Hence, the correct answer is C.

***Question 21:**

The value of is

**A.** 2 **B.** **C.** 0 **D.**

**Answer:**

Adding (1) and (2), we obtain

Hence, the correct answer is C.

**Miscellaneous Exercise**

***Question1:**

**Answer:**

Equating the coefficients of *x*^{2}, *x*, and constant term, we obtain

−*A* +* B *− *C* = 0

*B* + *C *= 0

*A* = 1

On solving these equations, we obtain

From equation (1), we obtain

***Question 2:**

**Answer:**

***Question 3:**

[Hint: Put]

**Answer:**

***Question 4:**

**Answer:**

***Question 5:**

**Answer:**

On dividing, we obtain

***Question 6:**

**Answer:**

Equating the coefficients of *x*^{2}, *x*, and constant term, we obtain

*A* + *B* = 0

*B *+ *C* = 5

9*A* + *C *= 0

On solving these equations, we obtain

From equation (1), we obtain

***Question 7:**

**Answer:**

Let *x *−* a *= *t*

⇒ *dx* = *dt*

***Question 8:**

**Answer:**

***Question 9:**

**Answer:**

Let sin *x* = *t*

⇒ cos *x dx* = *dt*

***Question 10:**

**Answer:**

***Question 11:**

**Answer:**

***Question 12:**

**Answer:**

Let *x*^{4 }=* t*

⇒ 4*x*^{3} *dx* = *dt*

***Question 13:**

**Answer:**

Let *ex* = *t*

⇒ *ex* *dx* = *dt*

***Question 14:**

**Answer:**

Equating the coefficients of *x*^{3}, *x*^{2}, *x*, and constant term, we obtain

*A* + *C* = 0

*B* + *D* = 0

4*A* + *C* = 0

4*B *+ *D* = 1

On solving these equations, we obtain

From equation (1), we obtain

***Question 15:**

**Answer:**

= cos^{3} *x* × sin *x*

Let cos *x* =* t*

⇒ −sin *x dx* =* dt*

***Question 16:**

**Answer:**

***Question 17:**

**Answer:**

***Question 18:**

**Answer:**

***Question 19:**

**Answer:**

….(i)

= taking

Let

∴ Integral

….(2)

From (1) and (2)

***Question 20:**

**Answer:**

***Question 21:**

**Answer:**

***Question 22:**

**Answer:**

Equating the coefficients of *x*^{2}, *x*,and constant term, we obtain

*A* + *C* = 1

3*A* + *B* + 2*C *= 1

2*A* + 2*B* + *C* = 1

On solving these equations, we obtain

*A* = −2, *B* = 1, and *C* = 3

From equation (1), we obtain

***Question 23:**

**Answer:**

***Question 24:**

**Answer:**

Integrating by parts, we obtain

***Question 25:**

**Answer:**

***Question 26:**

**Answer:**

When *x *= 0, *t *= 0 and when

***Question 27:**

**Answer:**

When and when

***Question 28:**

**Answer:**

When and when

As therefore, is an even function.

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

***Question 29:**

**Answer:**

***Question 30:**

**Answer:**

***Question 31:**

**Answer:**

From equation (1), we obtain

***Question 32:**

**Answer:**

Adding (1) and (2), we obtain

***Question 33:**

**Answer:**

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

***Question 34:**

**Answer:**

Equating the coefficients of *x*^{2}, *x*, 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

Hence, the given result is proved.

***Question 35:**

**Answer:**

Integrating by parts, we obtain

Hence, the given result is proved.

***Question 36:**

**Answer:**

Therefore, *f* (*x*) is an odd function.

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

Hence, the given result is proved.

***Question 37:**

**Answer:**

Hence, the given result is proved.

***Question 38:**

**Answer:**

Hence, the given result is proved.

***Question 39:**

**Answer:**

Integrating by parts, we obtain

Let 1 − *x*^{2} = *t*

⇒ −2*x* *dx* = *dt*

Hence, the given result is proved.

***Question 40:**

Evaluate as a limit of a sum.

**Answer:**

It is known that,

***Question 41:**

is equal to

**A.** **B.**

**C.** **D. **

**Answer:**

Hence, the correct answer is A.

***Question 42:**

is equal to

**A.** **B.**

**C.** **D. **

**Answer:**

Hence, the correct answer is B.

***Question 43:**

If then is equal to

**A.** **B.**

**C.** **D. **

**Answer:**

Hence, the correct answer is D.

***Question 44:**

The value of is

**A.** 1 **B.** 0 **C.** **D. **

**Answer:**

Adding (1) and (2), we obtain

Hence, the correct answer is B.

Conclusion Swastik Classes’ NCERT Solution for Class 12 Mathematics Chapter 7, “Integrals,” provides a comprehensive study material that helps students understand the fundamental concepts of integral calculus. The solutions offer step-by-step explanations and solved examples that help students develop a deeper understanding of the subject. The chapter covers various topics, including properties of integrals, methods of integration, integration using partial fractions, and integration using trigonometric identities. With the help of these solutions, students can improve their problem-solving skills and gain the confidence to tackle complex integral problems. Swastik Classes’ NCERT solutions are designed according to the latest CBSE syllabus, making them useful for students preparing for board exams or competitive exams like JEE and NEET. Overall, Swastik Classes’ NCERT Solution for Class 12 Mathematics Chapter 7 is an excellent resource for students who want to excel in mathematics and build a strong foundation in integral calculus.

### 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.