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

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Theorem

\(\ds \int_0^{\frac \pi 2} \sin^{2 n} x \rd x\) \(=\) \(\ds \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2\)
\(\ds \) \(=\) \(\ds \dfrac {1 \cdot 3 \cdot 5 \cdots \paren {2 n - 1} } {2 \cdot 4 \cdot 6 \cdots 2 n} \dfrac \pi 2\)

for $n \in \Z_{>0}$.


Proof

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

$\blacksquare$