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

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
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 1
Adding eqns.(1) and (2),
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 2

MP Board Solutions

Question 2.
Prove that \(\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\tan x}}=\frac{\pi}{12}\)? (NCERT)
Solution:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 3
Adding eqns.(1) and (2),
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 4

Question 3.
Evaluate \(\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} e^{x}(\log \sin x+\cot x) d x\)?
Solution:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 5

Question 4.
Evaluate \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}|2 \sin | x|+\cos | x|| d x\)?
Solution:
Let
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 6
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 6a

MP Board Solutions

Question 5.
Prove that:
\(\int_{0}^{\pi} \frac{x \tan x}{\sec x+\cos x} d x=\frac{\pi^{2}}{4}\)?
Solution:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 7
Adding eqns. (1) and (2)
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 8
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 8a

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:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 9
Adding eqns.(1) and (2)
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 10

Question 7.
Prove that:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 11
Solution:
Let
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 11
Adding eqns. (1) and (2)
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 13

Question 8.
Prove that:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 14
Solution:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 15
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 15a

Question 9.
Evaluate
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 16
Solution:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 17
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 17a
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 17b

MP Board Solutions

Question 10.
Prove that:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 18
Solution:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 19
Adding eqns.(1) and (2),
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 20

Question 11.
Evaluate
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 21
Solution:
Let
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 22
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 22a
When x = 0, then t = tan 0 = 0
and when x = π, then t = tan \(\frac { \pi }{ 2 } \) = ∞
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 23

MP Board Solutions

Question 12.
Prove that:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 24
Solution:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 25
Adding eqns. (1) and (2)
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 26
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 26a

Question 13.
Prove that:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 27
Solution:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 28
Adding eqns. (1) and (2)
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 29

MP Board Solutions

Question 14.
Prove that:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 30
Solution:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 31

Question 15.
Evaluate
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 32
Solution:
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 33
Adding eqns.(1) and (2)
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 34
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 34a

MP Board Solutions

Question 16.
Evaluate
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 34a
Solution:
Let
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 36
MP Board Class 12th Maths Important Questions Chapter 7B Definite Integral img 36a

MP Board Class 12 Maths Important Questions

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