Evaluate `int_(0)^(1)1/(sqrt(1-x^(2))"sin"^(-1)(2xsqrt(1-x^(2)))dx`.

Explanation

`I=int_(0)^(1)1/(sqrt(1-x^(2))"sin"^(-1)(2xsqrt(1-x^(2)))dx`
Put `x=sin theta`. When `x=0, theta=0` and when `x=1, theta=(pi)/2`.
`:.I=int_(0)^(pi//2)1/(sqrt(1-sin^(2)theta)"sin"^(-1)(2 sin theta cos theta)cos theta d theta`
`=int_(0)^(pi//2)sin^(-1)(sin 2 theta) d theta`
Put `2theta=1`
`:.I=1/2 int_(0)^(pi)sin^(-1)(sin t)dt`

From the graph `I=1/2` (area of triangle)
`=1/2xx1/2 (pi)/2 pi`
`=(pi^(2))/8`