## NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Miscellaneous Exercise

#### Page No 352:

#### Question 1:

#### 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

*e*^{x}=*t*⇒*e*^{x}*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:

Let I=∫sin-1x-cos-1xsin-1x+cos-1xdx

It is known that, sin-1x+cos-1x=π2

⇒I=∫π2-cos-1x-cos-1xπ2dx

=2π∫π2-2cos-1xdx

=2π.π2∫1.dx-4π∫cos-1xdx

=x-4π∫cos-1xdx …(1)

Let I1=∫cos-1x dx

Also, let x=t⇒dx=2 t dt

⇒I1=2∫cos-1t.t dt

=2cos-1t.t22-∫-11-t2.t22dt

=t2cos-1t+∫t21-t2dt

=t2cos-1t-∫1-t2-11-t2dt

=t2cos-1t-∫1-t2dt+∫11-t2dt

=t2cos-1t-t21-t2-12sin-1t+sin-1t

=t2cos-1t-t21-t2+12sin-1t

From equation (1), we obtain

I=x-4πt2cos-1t-t21-t2+12sin-1t =x-4πxcos-1x-x21-x+12sin-1x

=x-4πxπ2-sin-1x-x-x22+12sin-1x

#### 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

#### Page No 353:

#### Question 23:

#### Answer:

#### Question 24:

#### Answer:

Integrating by parts, we obtain

#### Question 25:

#### Answer:

#### Question 26:

#### Answer:

When

*x*= 0,*t*= 0 and#### 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.

#### Page No 354:

#### 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.**− 1

**D.**

#### Answer:

Adding (1) and (2), we obtain

Hence, the correct answer is B.

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