No, I don't think your way works. Before your substitution, you have x in the numerator so it doesn't look like the derivative of an arcsin. After your substitution, you have in the denominator, so again it doesn't look like the derivative of an arcsin.
But the integral is easy enough to do - you just substitute and it becomes a power of w.
- Hollywood
Your substitution looks ok, but you should have ended up with . There is NO point in bringing in extra square roots, as they would require more substitutions to be able to integrate.
What you should do is let and your integral becomes . Surely you can integrate that...
What hollywood and Prove It are saying is if it were simply then we can simply integrate. But your form of requires another substitution for the square root over u. It's the inner square root that is the problem here, not the outer.
-Dan