(\integral\_{0}^{\pi}{\sin^{5}\left( \frac{x}{2} \right)dx})... | MATH - SUMMATION OF SERIES BY DEFINITE INTEGRATION, GAMMA FUNCTION, LEIBNITZ'S RULE | SolveeIt

Question

$\int_{0}^{\pi}{\sin^{5}\left( \frac{x}{2} \right)dx}$ equals

$\frac{16}{15}$

$\frac{32}{15}$

$\frac{8}{15}$

$\frac{5}{6}$

Answer

$\frac{16}{15}$

Explanation

Solution

$\int_{0}^{\pi}{\sin^{5}\frac{2}{x}dx = 2\int_{0}^{\pi/2}{\sin^{5}tdt = 2.\frac{\Gamma\frac{6}{2}.\Gamma\frac{1}{2}}{2\Gamma\frac{7}{2}} = \frac{16}{15}}}$ .

Question: \(\int_{0}^{\pi}{\sin^{5}\left( \frac{x}{2} \right)dx}\) equals...

Solution