_0^ ^5 ( x 2 ) ,dx equals

MCQ

$\int_0^\pi {{{\sin }^5}\left( {\frac{x}{2}} \right)\,dx} $ equals

✓
$\frac{{16}}{{15}}$
B
$\frac{{32}}{{15}}$
C
$\frac{8}{{15}}$
D
$\frac{5}{6}$

✓

Answer

Correct option: A.

$\frac{{16}}{{15}}$

a
(a) $\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}}} } $.

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