I'm trying to evaluate $$\int_{0}^{\frac{\pi}{2}} \sqrt{\sin ^3 x \cos x} dx$$
This looks like an opportunity to isolate a $\sin x$ or $\cos x$ in order to use the reverse chain rule. However, the best I was able to do, in order to achieve this, was reduce the integral to $$\int_{0}^{\frac{\pi}{2}}\sqrt{\sqrt{1- \cos ^2 x} \cos x}\sin xdx$$
which then becomes, under the substitution $u=\cos x$
$$\int_{0}^{1} \sqrt[4]{1-u^2}\sqrt{u}du$$
But I am not quite sure what to do from here. I am not even sure if the integral can be found via elementary means, though Wolfram gives a closed form of $$\frac{\pi}{4\sqrt{2}}$$