# Definite Integral from 0 to Half Pi of Odd Power of Sine x/Proof 2

$\displaystyle \int_0^{\frac \pi 2} \sin^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$
 $\ds \int_0^{\pi/2} \sin^{2 n + 1} x \rd x$ $=$ $\ds \int_0^{\pi/2} \paren {\sin x}^{2 \paren {n + 1} - 1} \paren {\cos x}^{2 \paren {\frac 1 2} - 1} \rd x$ $\ds$ $=$ $\ds \frac 1 2 \map \Beta {n + 1, \frac 1 2}$ Definition 2 of Beta Function $\ds$ $=$ $\ds \frac {\map \Gamma {n + 1} \map \Gamma {\frac 1 2} } {2 \map \Gamma {n + \frac 3 2} }$ Definition 3 of Beta Function $\ds$ $=$ $\ds \frac {n! \sqrt \pi} {2 \paren {n + \frac 1 2} \map \Gamma {n + \frac 1 2} }$ Gamma Function Extends Factorial, Gamma Function of One Half, Gamma Difference Equation $\ds$ $=$ $\ds \frac {n! \sqrt \pi} {2 n + 1} \times \frac {2^{2 n} n!} {\paren {2 n!} \sqrt \pi}$ Gamma Function of Positive Half-Integer $\ds$ $=$ $\ds \frac {\paren {2^n n!}^2 } {\paren {2 n + 1}!}$
$\blacksquare$