Hey tempq1.
Try using a substitution u = 2x instead of arcsin(2x) and then see what the form of the integral is (and after that consider integrating by parts to get a cancellation).
We know that:
So for
So now:
For:
Let
Check:
http://www.wolframalpha.com/input/?i=differentiation+2*x^(3%2F2)*tan^(-1)(x^(1%2F2))%2F3+%2B+ln(1%2Bx)%2F3-(1%2Bx)%2F3+%2B+C