Evaluate: `intxsin^(-1)x\ dx`

Explanation

`I = int xsin^-1x dx`
Here, we will integrate by parts and `sin^-1x` will be first function.
`:. I = sin^-1x int xdx - int d/dx(sin^-1x) int xdx`
`=>I = sin^-1x(x^2/2) - int 1/sqrt(1-x^2)*x^2/2dx`
`=>I = (x^2/2)sin^-1x + 1/2int (-x^2)/sqrt(1-x^2)dx`
`=>I = (x^2/2)sin^-1x + 1/2int (1-x^2-1)/sqrt(1-x^2)dx`
`=>I = (x^2/2)sin^-1x + 1/2int (1-x^2)/sqrt(1-x^2)dx -1/2 int 1/sqrt(1-x^2)dx`
`=>I = (x^2/2)sin^-1x + 1/2int sqrt(1-x^2)dx -1/2 int 1/sqrt(1-x^2)dx`
Using the integral formula `int sqrt(a^2-x^2)dx = (xsqrt(a^2-x^2))/2+a^2/2sin^-1(x/a)+c`, we have,
`=>I = (x^2/2)sin^-1x + 1/2(1/2 (xsqrt(1-x^2)+sin^-1x)-1/2 sin^-1x +c`
`=>I = (x^2/2)sin^-1x + (xsqrt(1-x^2))/4-1/4 sin^-1x +c`