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

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Theorem

$\ds \int_0^{\frac \pi 2} \cos^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$


Proof

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

$\blacksquare$