Question

The correct evaluation of $\int\limits^{\pi/2}_{0} \left|sin^{4} x\right|dx$ 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)

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