Calculate $\int_{0}^{\pi/2} (\sin(2x))^5 dx$

Question

I want to calculate the following integral:
$$\int_{0}^{\pi/2} (\sin(2x))^5 dx$$I tried integration by parts but it didn't work for me. Suggestions?

Answer 1

How about this

\begin{eqnarray} \mathcal I &=& \int_0^{\pi/2}(\sin 2x)^5 dx=\\ &=&32\int_0^{\pi/2}\sin^5x \cos^5xdx=\\ &=&32\int_0^{\pi/2}\sin^5x(1-\sin^2x)^2\cos x dx=\\ &\stackrel{t=\sin x}{=}&32\int_0^1t^5(1-t^2)^2dt. \end{eqnarray}