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The value of int 0 ^ 1 x ^ 2 (...

The value of ` int _ 0 ^ 1 x ^ 2 ( 1 - x ^ 2 ) ^ (3//2 ) dx ` is

A

`1/32`

B

`pi/8`

C

`pi/16`

D

`pi/32`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `l = int_(0)^(1) x^(2)(1-x^(2))^(3//2)dx` [let `x = sin theta rArr dx = cos theta d theta`]
`l = int_(0)^(pi//2)sin^(2)theta (cos^(2)theta)^(3//2).cos theta d theta`
`= int_(0)^(pi//2)sin^(2)theta.cos^(4)theta d theta`
By Gamma function,

`= (3.sqrt(pi).sqrt(pi))/(2.2.2.2.3.2)=(pi)/(32)`
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