`intx sin^(-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`

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