Question

$\stackrel{2\pi }{\underset{0}{\int }}{\sin }^5\left(\frac{x}{4}\right)dx$ is equal to

• A. $\frac{8}{15}$
• B. $\frac{16}{15}$
• C. $\frac{32}{15}$
• D. $0$

Answer 1

The correct option is C $\frac{32}{15}$

Solution:

We have, $\stackrel{2\pi }{\underset{0}{\int }}{\sin }^5\left(\frac{x}{4}\right)dx$

Put, $\frac{x}{4}=t⟹dx=4dt$

$\therefore \stackrel{\frac{\pi }{2}}{\underset{0}{\int }}{\sin }^{5}t.4dt=$$4\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}{\sin }^{5}tdt$

By Walli's formula

$\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}{\sin }^{n}xdx=$$\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}{\cos }^{n}xdx$

$=\frac{n-1}{n}.\frac{n-3}{n-2}.\frac{n-5}{n-4}....\frac{2}{3}$ when $n$ is odd.

$=\frac{n-1}{n}.\frac{n-3}{n-2}.\frac{n-5}{n-4}....\frac{3}{4}.\frac{1}{2}.\frac{\pi }{2}$ when $n$ is even.

$=4.\frac{4.2}{5.3}=\frac{32}{15}$

Hence, C is the correct option.