Definite Integral from 0 to Half Pi of Odd Power of Sine x/Proof 2
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Theorem
\(\ds \int_0^{\frac \pi 2} \sin^{2 n + 1} x \rd x\) | \(=\) | \(\ds \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 \cdot 4 \cdot 6 \cdots 2 n} {3 \cdot 5 \cdot 7 \cdots \paren {2 n + 1} }\) |
for $n \in \Z_{>0}$.
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {2^n n!}^2 } {\paren {2 n + 1}!}\) |
$\blacksquare$