Updated By SaraNextGen

On April 21, 2024, 11:35 AM

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 3x as second function and integrating by parts, we obtain

Question 3:

Answer:

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

Answer:

Let 

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

Question 5:

x log 2x

Answer:

Let 

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

Question 6:

x² log x

Answer:

Let 

Taking log x as first function and x² 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 sec²x 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 = I₁ + I₂ … (1)

Where,  and 

Taking log x as first function and x² 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  ⇒ 

 = 2θ

⇒ 

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.