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

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 1

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

Then:

 $\ds I_{2 n + 1}$ $=$ $\ds \frac {2 n} {2 n + 1} I_{2 n - 1}$ Reduction Formula for Definite Integral of Power of Sine $\ds$ $=$ $\ds \frac {2 n \paren {2 n - 2} } {\paren {2 n + 1} \paren {2 n - 1} } I_{2 n - 3}$ Reduction Formula for Definite Integral of Power of Sine again $\ds$ $=$ $\ds \frac {2 n \paren {2 n - 2} \dotsm 2} {\paren {2 n + 1} \paren {2 n - 1} \dotsm 3} I_1$ Reduction Formula for Definite Integral of Power of Sine until the end $\ds$ $=$ $\ds \frac {2 n \paren {2 n - 2} \dotsm 2} {\paren {2 n + 1} \paren {2 n - 1} \dotsm 3} \int_0^{\pi / 2} \sin x \rd x$ Definition of $I_n$ $\ds$ $=$ $\ds \frac {2 n \paren {2 n - 2} \dotsm 2} {\paren {2 n + 1} \paren {2 n - 1} \dotsm 3} \bigintlimits {-\cos x} 0 {\pi / 2}$ Primitive of Sine Function $\ds$ $=$ $\ds \frac {2 n \paren {2 n - 2} \dotsm 2} {\paren {2 n + 1} \paren {2 n - 1} \dotsm 3} \paren {0 - \paren {-1} }$ Cosine of Right Angle and Cosine of Zero is One $\ds$ $=$ $\ds \frac {2 n \paren {2 n - 2} \dotsm 2} {\paren {2 n + 1} \paren {2 n - 1} \dotsm 3}$ $\ds$ $=$ $\ds \frac {\paren {2 n}^2 \paren {2 n - 2}^2 \dotsm 2^2} {\paren {2 n + 1} \paren {2 n} \paren {2 n - 1} \paren {2 n - 2} \paren {2 n - 3} \dotsm 3 \cdot 2}$ multiplying top and bottom by top $\ds$ $=$ $\ds \frac {\paren {2 n}^2 n^2 \paren {n - 1}^2 \dotsm 1^2} {\paren {2 n + 1} \paren {2 n} \paren {2 n - 1} \paren {2 n - 2} \paren {2 n - 3} \dotsm 3 \cdot 2}$ extracting factor of $\paren {2^n}^2$ from the top $\ds$ $=$ $\ds \frac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$ Definition of Factorial

$\blacksquare$

Proof 2

 $\ds \int_0^{\pi/2} \sin^{2 n + 1} x \rd x$ $=$ $\ds \int_0^{\pi/2} \paren {\sin x}^{2 \paren {n + 1} - 1} \paren {\cos x}^{2 \paren {\frac 1 2} - 1} \rd x$ $\ds$ $=$ $\ds \frac 1 2 \map \Beta {n + 1, \frac 1 2}$ 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 $\ds$ $=$ $\ds \frac {\paren {2^n n!}^2 } {\paren {2 n + 1}!}$

$\blacksquare$