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

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Theorem

Let $n \in \Z_{> 0}$ be a positive integer.


Then:

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


Proof 1

Let $I_n = \displaystyle \int_0^{\frac \pi 2} \sin^n x \rd x$.


Then:

\(\displaystyle I_{2 n}\) \(=\) \(\displaystyle \frac {2 n - 1} {2 n} I_{2 n - 2}\) Reduction Formula for Definite Integral of Power of Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {2 n - 1} \paren {2 n - 3} } {2 n \paren {2 n - 2} } I_{2 n - 4}\) Reduction Formula for Definite Integral of Power of Sine again
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {2 n - 1} \paren {2 n - 3} \cdots 1} {2 n \paren {2 n - 2} \cdots 2} I_0\) Reduction Formula for Definite Integral of Power of Sine until the end
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {2 n - 1} \paren {2 n - 3} \cdots 1} {2 n \paren {2 n - 2} \cdots 2} \int_0^{\pi / 2} \rd x\) Definition of $I_n$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {2 n - 1} \paren {2 n - 3} \cdots 1} {2 n \paren {2 n - 2} \cdots 2} \bigintlimits x 0 {\pi / 2}\) Integral of Constant
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {2 n - 1} \paren {2 n - 3} \cdots 1} {2 n \paren {2 n - 2} \cdots 2} \frac \pi 2\) Integral of Constant
\(\displaystyle \) \(=\) \(\displaystyle \frac {2n \paren {2 n - 1} \paren {2 n - 2} \paren {2 n - 3} \cdots 2 \cdot 1} {\paren {2 n}^2 \paren {2 n - 2}^2 \cdots 2^2} \frac \pi 2\) multiplying top and bottom by bottom
\(\displaystyle \) \(=\) \(\displaystyle \frac {2n \paren {2 n - 1} \paren {2 n - 2} \paren {2 n - 3} \cdots 2 \cdot 1} {\paren {2^n}^2 n^2 \paren {n - 1}^2 \cdots 1^2} \frac \pi 2\) extracting factor of $\paren {2^n}^2$ from the bottom
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {2 n}!} {\paren {2^n}^2 \paren {n!}^2} \frac \pi 2\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {2 n}!} {\paren {2^n n!}^2} \frac \pi 2\) rearranging

$\blacksquare$


Proof 2

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

$\blacksquare$


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