Definite Integral from 0 to Half Pi of Odd Power of Sine x

Theorem
Let $n \in \Z_{\ne 0}$ be a positive integer.

Then:
 * $\displaystyle \int_0^{\frac \pi 2} \sin^{2 n + 1} x \ \mathrm d x = \dfrac {\left({2^n n!}\right)^2} {\left({2 n + 1}\right)!}$

Proof
Let $I_n = \displaystyle \int_0^{\frac \pi 2} \sin^n x \ \mathrm d x$.

Then: