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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {2 n}!} {\paren {2^n n!}^2} \frac \pi 2\) |
$\blacksquare$