Add new solution

In my opinion, using substitution makes the answer more simple. Also, I guess many students first think about substitution $u=\cos x$ for

$\int _{0}^{\pi }\sin ^{5}xdx=\int _{0}^{\pi }\sin ^{4}x\sin xdx-\int _{0}^{\pi }(1+\cos ^{2}x)^{2}\sin xdx$,

instead of using integration by parts.

If you agree, can you move the current solution to the second solution, and write the first hint and solution based on substitution?