## MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral

### Definite Integral Important Questions

**Definite Integral Long Answer Type Questions – II**

Question 1.

Evaluate \(\int_{0}^{4 a} \frac{f(x)}{f(x)+f(4 a-x)} d x\)?

Solution:

Let

Adding eqns.(1) and (2),

Question 2.

Prove that \(\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\tan x}}=\frac{\pi}{12}\)? (NCERT)

Solution:

Adding eqns.(1) and (2),

Question 3.

Evaluate \(\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} e^{x}(\log \sin x+\cot x) d x\)?

Solution:

Question 4.

Evaluate \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}|2 \sin | x|+\cos | x|| d x\)?

Solution:

Let

Question 5.

Prove that:

\(\int_{0}^{\pi} \frac{x \tan x}{\sec x+\cos x} d x=\frac{\pi^{2}}{4}\)?

Solution:

Adding eqns. (1) and (2)

Question 6.

Prove that:

\(\int_{0}^{\pi / 2} \frac{x \sin x \cos x}{\cos ^{4} x+\sin ^{4} x} d x=\frac{\pi^{2}}{16}\)?

Solution:

Adding eqns.(1) and (2)

Question 7.

Prove that:

Solution:

Let

Adding eqns. (1) and (2)

Question 8.

Prove that:

Solution:

Question 9.

Evaluate

Solution:

Question 10.

Prove that:

Solution:

Adding eqns.(1) and (2),

Question 11.

Evaluate

Solution:

Let

When x = 0, then t = tan 0 = 0

and when x = π, then t = tan \(\frac { \pi }{ 2 } \) = ∞

Question 12.

Prove that:

Solution:

Adding eqns. (1) and (2)

Question 13.

Prove that:

Solution:

Adding eqns. (1) and (2)

Question 14.

Prove that:

Solution:

Question 15.

Evaluate

Solution:

Adding eqns.(1) and (2)

Question 16.

Evaluate

Solution:

Let