Definite Integral from 0 to Half Pi of Odd Power of Cosine x/Proof 2

Theorem

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

$\ds \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x$

$=$

$\ds \int_0^{\frac \pi 2} \paren {\sin x}^{\frac 2 2 - 1} \paren {\cos x}^{2 \paren {n + 1} - 1} \rd x$

$\ds $

$=$

$\ds \frac 1 2 \map \Beta {\frac 1 2, n + 1}$

Definition 2 of Beta Function

$\ds $

$=$

$\ds \frac 1 2 \cdot \frac {\map \Gamma {n + 1} \map \Gamma {\frac 1 2} } {\map \Gamma {n + 1 + \frac 1 2} }$

Definition 3 of Beta Function

$\ds $

$=$

$\ds \frac 1 2 \cdot \frac {n! \sqrt \pi} {\map \Gamma {n + 1 + \frac 1 2} }$

$\ds $

$=$

$\ds \frac 1 2 \cdot n! \sqrt \pi \cdot \frac{2^{2 n + 2} \paren {n + 1}!} {\paren {2 n + 2}! \sqrt \pi}$

$\ds $

$=$

$\ds \frac {n! \cdot 2^{2 n + 1} \paren {n + 1}!} {\paren {2 n + 2} \paren {2 n + 1}!}$

$\ds $

$=$

$\ds \frac 2 2 \cdot \frac {n! \cdot 2^{2 n} \paren {n + 1} n!} {\paren {n + 1} \paren {2 n + 1}!}$

$\ds $

$=$

$\ds \frac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$

$\blacksquare$