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

#### Question 1:

x sin x

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

#### Question 2:

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

#### Question 3:

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

#### Question 4:

x logx

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

#### Question 5:

x log 2x

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

#### Question 6:

xlog x

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

#### Question 7:

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

#### Question 8:

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

#### Question 9:

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

#### Question 10:

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

#### Question 11:

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

#### Question 12:

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

#### Question 13:

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

#### Question 14:

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:

Let
Let I = I1 + I2 … (1)
Where, and
Taking log x as first function and xas 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:

Let
Let
⇒
∴
It is known that,

#### Question 17:

Let
Let  ⇒
It is known that,

#### Question 18:

Let  ⇒
It is known that,
From equation (1), we obtain

#### Question 19:

Also, let  ⇒
It is known that,

#### Question 20:

Let  ⇒
It is known that,

#### Question 21:

Let
Integrating by parts, we obtain
Again integrating by parts, we obtain

#### Question 22:

Let ⇒
= 2θ
⇒
Integrating by parts, we obtain

#### Question 23:

equals

Let
Also, let  ⇒
Hence, the correct answer is A.

equals