Evaluate the following integrals:
`intxsin^(-1)xdx`

Answer

`(1)/(2)x^(2)sin^(-1)x+(1)/(4)xsqrt(1-x^(2))-(1)/(4)sin^(-1)x+C`

Explanation

Integrate by parts, taking `sin^(-1)x` as 1st function and x as 2nd.
`I=(sin^(-1)x).(x^(2))/(2)-int(1)/(1-x^(2))*(x^(2))/(2)x^(2)(sin^(-1)x)+(1)/(2)*int((1-x^(2))-1)/(sqrt(1-x^(2)))dx`
`=(1)/(2)x^(2)(sin^(-1)x)+(1)/(2)*intsqrt(1-x^(2))dx-(1)/(2)int(dx)/(sqrt(1-x^(2)))`
`=(1)/(2)x^(2)(sin^(-1)x)+(1)/(2)*{:[(xsqrt(1-x^(2)))/(2)+(1)/(2)sin^(-1)x]:}-(1)/(2)sin^(-1)x+C`.