Question

The value of the integral ${\int }_{-\pi /4}^{\pi /4}{\sin }^{-4}xdx$ is

• A. -8/3
• B. 3/2
• C. 8/3
• D. none oft hese

Answer 1

Answer: (A)

Let $I=\underset{-\pi /4}{\overset{\pi /4}{\int }}si{n}^{-4}xdx=\underset{-\pi /4}{\overset{\pi /4}{\int }}\frac{dx}{si{n}^{4}x}$
$=\underset{-\pi /4}{\overset{\pi /4}{\int }}cose{c}^{4}xdx$
Put cot $x=t\therefore -cose{c}^{2}xdx=dt$.
$\therefore I=-\underset{-1}{\overset{1}{\int }}\left(1+t^2\right)dt$
$=-2\underset{0}{\overset{1}{\int }}\left(1+t^2\right)dt=-2{\left[t+\frac{t^3}{3}\right]}_0^1$
$=-\frac{8}{3}$