## EXERCISE 1.6

#### Page No 327:

#### Question 1:

*x* sin *x*

#### Answer:

Let *I* =

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

#### Question 2:

#### Answer:

Let *I* =

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 *e*^{x} 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

I=log x 2∫xdx-∫ddxlog x 2∫xdxdx=x22log x 2-∫2log x .1x.x22dx=x22log x 2-∫xlog x dx

Again integrating by parts, we obtain

I = x22logx 2-log x ∫x dx-∫ddxlog x ∫x dxdx=x22logx 2-x22log x -∫1x.x22dx=x22logx 2-x22log x +12∫x dx=x22logx 2-x22log x +x24+C

#### 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

#### Page No 328:

#### 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.