Evaluate: `int_0^(1/(sqrt(2)))(sin^(-1)x)/((1-x^2)sqrt(1-x^2))dx`

Answer

`(pi)/4-1/2 log2`

Explanation

Put `x=sin theta`.So `dx=cos theta d theta`
When `x=0, theta=0`, when `x=1/(sqrt(2)),theta=(pi)/4`
`:.` Given integral
`=int_(0)^(pi//4)(sin^(-1)(sin theta)cos theta d theta)/((1-sin^(2) theta)^(3//2))`
`=int_(0)^(pi//4)(theta cos theta)/(cos^(3) theta) d theta= int_(1)^(pi//4) underset(I)(theta).underset(II)(sec^(2)) theta d theta`
`=|theta tan theta |_(0)^(pi//4)-int_(0)^(pi//4)1.tan d theta`
`=(pi)/4 "tan" (pi)/4+log cos theta|_(0)^(pi//4)`
`=(pi)/4+"log cos"(pi)/4-"log cos"=(pi)/4+"log"1/(sqrt(2))`
`=(pi)/4+log1-log(2)^(1//2)=(pi)/4-1/2 log 2`