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

## 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 $\displaystyle$ $=$ $\displaystyle \frac {\paren {2 n}!} {\paren {2^n n!}^2} \frac \pi 2$

$\blacksquare$