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

Theorem

\(\ds \int_0^{\frac \pi 2} \sin^{2 n} x \rd x\)

\(=\)

\(\ds \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2\)

\(\ds \)

\(=\)

\(\ds \dfrac {1 \cdot 3 \cdot 5 \cdots \paren {2 n - 1} } {2 \cdot 4 \cdot 6 \cdots 2 n} \dfrac \pi 2\)

for $n \in \Z_{>0}$.

Proof

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

Then:

\(\ds I_{2 n}\)

\(=\)

\(\ds \frac {2 n - 1} {2 n} I_{2 n - 2}\)

Reduction Formula for Definite Integral of Power of Sine

\(\ds \)

\(=\)

\(\ds \frac {\paren {2 n - 1} \paren {2 n - 3} } {2 n \paren {2 n - 2} } I_{2 n - 4}\)

Reduction Formula for Definite Integral of Power of Sine again

\(\ds \)

\(=\)

\(\ds \frac {\paren {2 n - 1} \paren {2 n - 3} \cdots 1} {2 n \paren {2 n - 2} \cdots 2} I_0\)

Reduction Formula for Definite Integral of Power of Sine until the end

\(\ds \)

\(=\)

\(\ds \frac {\paren {2 n - 1} \paren {2 n - 3} \cdots 1} {2 n \paren {2 n - 2} \cdots 2} \int_0^{\pi / 2} \rd x\)

Definition of $I_n$

\(\ds \)

\(=\)

\(\ds \frac {\paren {2 n - 1} \paren {2 n - 3} \cdots 1} {2 n \paren {2 n - 2} \cdots 2} \bigintlimits x 0 {\pi / 2}\)

Integral of Constant

\(\ds \)

\(=\)

\(\ds \frac {\paren {2 n - 1} \paren {2 n - 3} \cdots 1} {2 n \paren {2 n - 2} \cdots 2} \frac \pi 2\)

Integral of Constant

\(\ds \)

\(=\)

\(\ds \frac {2n \paren {2 n - 1} \paren {2 n - 2} \paren {2 n - 3} \cdots 2 \cdot 1} {\paren {2 n}^2 \paren {2 n - 2}^2 \cdots 2^2} \frac \pi 2\)

multiplying top and bottom by bottom

\(\ds \)

\(=\)

\(\ds \frac {2n \paren {2 n - 1} \paren {2 n - 2} \paren {2 n - 3} \cdots 2 \cdot 1} {\paren {2^n}^2 n^2 \paren {n - 1}^2 \cdots 1^2} \frac \pi 2\)

extracting factor of $\paren {2^n}^2$ from the bottom

\(\ds \)

\(=\)

\(\ds \frac {\paren {2 n}!} {\paren {2^n}^2 \paren {n!}^2} \frac \pi 2\)

Definition of Factorial

\(\ds \)

\(=\)

\(\ds \frac {\paren {2 n}!} {\paren {2^n n!}^2} \frac \pi 2\)

rearranging

$\blacksquare$