Question

The correct evaluation of ${\int }_0^{\pi }|si{n}^{4}x|dx$ is [MP PET 1993]

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

Answer 1

The correct option is D $\frac{3\pi }{8}$

${\int }_0^{\pi }|si{n}^{4}x|dx=2{\int }_0^{\frac{\pi }{2}}si{n}^{4}xdx$
Applying gamma function,
$2{\int }_0^{\frac{\pi }{2}}si{n}^{4}xdx=2\frac{T\left(\frac{5}{2}\right).T\left(\frac{1}{2}\right)}{2.T\left(\frac{6}{2}\right)}=\frac{3\pi }{8}$