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

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Theorem

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


Proof

\(\ds \int_0^{\pi/2} \sin^{2 n + 1} x \rd x\) \(=\) \(\ds \int_0^{\pi/2} \paren {\sin x}^{2 \paren {n + 1} - 1} \paren {\cos x}^{2 \paren {\frac 1 2} - 1} \rd x\)
\(\ds \) \(=\) \(\ds \frac 1 2 \map \Beta {n + 1, \frac 1 2}\) 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
\(\ds \) \(=\) \(\ds \frac {\paren {2^n n!}^2 } {\paren {2 n + 1}!}\)

$\blacksquare$