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

Explanation

Let `I=int_(0)^(1//2)(xsin^(-1)x)/(sqrt(1-x^(2)))dx`
Put `x=sin theta` and `dx=cos theta d theta`
Also, when `x=0, theta=0`, and when `x=1//2,theta=pi//6`.
Thus, `I=int_(0)^(pi//6)(sin theta sin^(-1)(sin theta))/(sqrt(1-sin^(2)theta)) cos theta d theta =int_(0)^(pi//6) theta sin theta d theta`
Integrating by parts, we get
`I=[theta(-cos theta)]_(0)^(pi//6)+int_(0)^(pi//6)1 cos theta d theta`
`=[-theta cos theta +sin theta ]_(0)^(pi//6)`
`=(-pi)/6(sqrt(3))/2+1/2=(6-pisqrt(3))/12`