Evaluate `int_(0)^(1//2)(xsin^(-1)x)/(sqrt(1-x^(2)))dx`.

Answer

`(-(sqrt(3))/(12)pi+(1)/(2))`

Explanation

` = int _(0)^(1//2) (x sin ^(-1))/(sqrt(1-x^(2)))dx " put" sin^(-1) x = theta rArr x = sin theta`
`rArr dx = cos theta d theta`
` :. I = int _(0)^(pi//6) (thetas sin theta)/(sqrt(1- sin ^(2)theta))* cos theta d theta = int _(0) ^(pi//6) theta sin d theta`
`=[- theta cos theta]_(0)^(pi//6) + int_(0)^(pi//6)cos theta d theta`
` =(-(pi)/(6)"cos"(pi)/(6)+0)+("sin"(pi)/(6)-sin0)=-(sqrt(3)pi)/(12)+(1)/(2)`