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

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Theorem

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

Then:

$\displaystyle \int_0^{\frac \pi 2} \sin^{2 n + 1} x \rd x = \dfrac {\left({2^n n!}\right)^2} {\left({2 n + 1}\right)!}$


Proof

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


Then:

\(\displaystyle I_{2 n + 1}\) \(=\) \(\displaystyle \frac {2 n} {2 n + 1} I_{2 n - 1}\) $\quad$ Reduction Formula for Definite Integral of Power of Sine $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 n \left({2 n - 2}\right)} {\left({2 n + 1}\right) \left({2 n - 1}\right)} I_{2 n - 3}\) $\quad$ Reduction Formula for Definite Integral of Power of Sine again $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 n \left({2 n - 2}\right) \cdots 2} {\left({2 n + 1}\right) \left({2 n - 1}\right) \cdots 3} I_1\) $\quad$ Reduction Formula for Definite Integral of Power of Sine until the end $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 n \left({2 n - 2}\right) \cdots 2} {\left({2 n + 1}\right) \left({2 n - 1}\right) \cdots 3} \int_0^{\pi / 2} \sin x \rd x\) $\quad$ Definition of $I_n$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 n \left({2 n - 2}\right) \cdots 2} {\left({2 n + 1}\right) \left({2 n - 1}\right) \cdots 3} \left[{-\cos x}\right]_0^{\pi / 2}\) $\quad$ Primitive of Sine Function $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 n \left({2 n - 2}\right) \cdots 2} {\left({2 n + 1}\right) \left({2 n - 1}\right) \cdots 3} \left({0 - \left({-1}\right)}\right)\) $\quad$ Cosine of Right Angle and Cosine of Zero is One $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 n \left({2 n - 2}\right) \cdots 2} {\left({2 n + 1}\right) \left({2 n - 1}\right) \cdots 3}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\left({2 n}\right)^2 \left({2 n - 2}\right)^2 \cdots 2^2} {\left({2 n + 1}\right) \left({2 n}\right)\left({2 n - 1}\right) \left({2 n - 2}\right) \left({2 n - 3}\right) \cdots 3 \cdot 2}\) $\quad$ multiplying top and bottom by top $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\left({2^n}\right)^2 n^2 \left({n - 1}\right)^2 \cdots 1^2} {\left({2 n + 1}\right) \left({2 n}\right)\left({2 n - 1}\right) \left({2 n - 2}\right) \left({2 n - 3}\right) \cdots 3 \cdot 2}\) $\quad$ extracting factor of $\left({2^n}\right)^2$ from the top $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\left({2^n n!}\right)^2} {\left({2 n + 1}\right)!}\) $\quad$ Definition of Factorial $\quad$

$\blacksquare$


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