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

Options

A. ` ( 1 )/( 32 ) `
B. ` ( pi ) /( 8 ) `
C. ` ( pi ) /( 16 ) `
D. ` ( pi ) /( 32 ) `

Answer

D

Explanation

Let ` I = int _ 0 ^ 1 x ^ 2 ( 1 - x ^ 2) ^ ( 3//2 ) dx `
( let ` x = sin theta rArr dx = cos theta d theta ` )
` I= 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,
` I = ( sqrt ( ( 2 + 1 ) /( 2 ) ) sqrt ( ( 4 + 1 )/ ( 2 )) ) /( 2 sqrt (( 2 + 4 + 2 ) / ( 2 )) ) = ( sqrt ( ( 3 )/ (2 ) ) sqrt ( ( 5 )/ (2 )) )/ ( 2sqrt 4 ) `
` = ( ( 1 )/ ( 2 ) sqrt (( 1 )/ (2 ) ) * (3 )/ (2 ) * ( 1 ) / ( 2 ) * sqrt ( ( 1 )/ (2 )))/ (2 * 3 * 2 ) `
` = ( 3 * sqrt ( pi ) * sqrt ( pi) ) /( 2 * 2 * 2 * 2 * 3 * 2 ) = ( pi ) /( 32 ) `