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  4. The value of the integral displaystyle∫-π/4π/4 sin-4x dx is

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Q. The value of the integral $\displaystyle\int_{-\pi/4}^{\pi/4}\sin^{-4}x\,dx$ is

Integrals

Solution:

Let $I=\int\limits_{-\pi /4}^{\pi /4} sin^{-4} x \,dx=\int\limits_{-\pi /4}^{\pi/ 4} \frac{dx}{sin^{4}\,x}$
$=\int\limits_{-\pi /4}^{\pi /4}cosec^{4} \,x\,dx$
Put cot $x = t \therefore -cosec^{2}\,x\,dx=dt$.
$\therefore I=-\int\limits_{-1}^{1}\left(1+t^{2}\right)dt$
$=-2 \int\limits_{0}^{1}\left(1+t^{2}\right)dt=-2\left[t+\frac{t^{3}}{3}\right]_{0}^{1}$
$=-\frac{8}{3}$