# Definite Integral from 0 to Half Pi of Odd Power of Sine x/Proof 1

## Theorem

$\displaystyle \int_0^{\frac \pi 2} \sin^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$

## Proof

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

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 $I_n$ $\ds$ $=$ $\ds \frac {2 n \paren {2 n - 2} \dotsm 2} {\paren {2 n + 1} \paren {2 n - 1} \dotsm 3} \intlimits {-\cos x} 0 {\pi / 2}$ 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 $\left({2^n}\right)^2$ from the top $\ds$ $=$ $\ds \frac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$ Definition of Factorial

$\blacksquare$