Question

The correct evaluation of $\underset{0}{\overset{\pi /2}{\int }}\left|si{n}^{4}x\right|dx$ is

• A. $\frac{8\pi }{3}$
• B. $\frac{2\pi }{3}$
• C. $\frac{4\pi }{3}$
• D. $\frac{3\pi }{8}$

Answer 1

Answer: (D)

$\underset{0}{\overset{\pi }{\int }}\left|si{n}^{4}x\right|dx=\underset{0}{\overset{\pi /2}{\int }}si{n}^{4}xdx$
Applying gamma function,
$2\underset{0}{\overset{\pi /2}{\int }}si{n}^{4}xdx=2\frac{\Gamma \left(5/2\right).\Gamma \left(1/2\right)}{2.\Gamma \left(6/2\right)}=\frac{3\pi }{8}$
Aliter : Using Walli’s formula
$=2\left[\frac{3.1}{4.2}\right]\frac{\pi }{2}=\frac{3\pi }{8}$