## NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Ex 7.6

#### 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.

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