The correct evaluation of _0^ | , ^4 x , | ,dx is - Vidyadip

MCQ

The correct evaluation of $\int_0^\pi {\left| {\,{{\sin }^4}x\,} \right|\,dx} $ is

A
$\frac{{8\pi }}{3}$
B
$\frac{{2\pi }}{3}$
C
$\frac{{4\pi }}{3}$
✓
$\frac{{3\pi }}{8}$

✓

Answer

Correct option: D.

$\frac{{3\pi }}{8}$

d
(d) $\int_0^\pi {{\rm{ }}\left| {{\rm{ }}{{\sin }^4}x{\rm{ }}} \right|{\rm{ }}dx = 2\int_0^{\pi /2} {{{\sin }^4}x\,dx} } $

Applying gamma function,

$2\int_0^{\pi /2} {{{\sin }^4}x\,dx} = 2\frac{{\Gamma (5/2).\Gamma (1/2)}}{{2.\Gamma (6/2)}} = \frac{{3\pi }}{8}$.

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