Evaluate: `int(sin^(-1)x)/((1-x^2)^(3setminus2))\ dx`

Explanation

Put `x=sint" so that "dx=cosandt=sin^(-1)x`.
`:.int(sin^(-1)x)/((1-x^(2))^(3//2))dx=int(tcost)/((1-sin^(2)t)^(3//2))dtint(tcost)/(cos^(3)t)dt`
`=intsec^(2)tdt`
`=t*(tant)-int1*tantdt" "` [integrating by parts]
`=t*(tant)+log|cost|+C`
`=(sin^(-1)x)*(x)/(sqrt(1-x^(2)))+log|sqrt(1-x^(2))|+C`.
`[becausecost=sqrt(1-x^(2))andtant=(x)/(sqrt(1-x^(2)))]`
`=(x(sin^(-1)x))/(sqrt(1-x^(2)))+(1)/(2)log|(1-x^(2))|+C`