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

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

Then:
 * $\displaystyle \int_0^{\frac \pi 2} \sin^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$