That looks much too complicated! Whoever gave you this problem clearly expects that you have previously learned that the derivative of \(\displaystyle y= \arcsin(x)\) is \(\displaystyle y'= \frac{1}{\sqrt{1- x^2}}\) so an obvious substitution is \(\displaystyle u= \arcsin(x)\). Then \(\displaystyle du= \frac{dx}{\sqrt{1- x^2}}\) and the integral becomes \(\displaystyle \int\frac{du}{u}\) which should be easy.
