Mathematics

If $$\displaystyle \int_{0}^{\infty}{\dfrac{\sin x}{x}dx}=\dfrac{\pi}{2}$$, then $$\displaystyle \int_{0}^{\infty}{\dfrac{\sin^{3}x}{x}dx}$$ is equal to


ANSWER

$$\pi/2$$


SOLUTION
We know, $$\sin ^{3} x=\dfrac{3}{4} \sin x-\dfrac{1}{4} \sin 3 x\\$$
$$\displaystyle \Rightarrow \int_{0}^{\infty} \dfrac{\sin ^{3} x}{x} d x\\$$
$$\displaystyle=\dfrac{3}{4} \int_{0}^{\infty} \dfrac{\sin x}{x} d x-\dfrac{1}{4} \int_{0}^{\infty} \dfrac{\sin 3 x}{x} d x\\$$
$$\displaystyle=\dfrac{3}{4} \int_{0}^{\infty} \dfrac{\sin x}{x} d x-\dfrac{1}{4} \int_{0}^{\infty} \dfrac{\sin u}{u} d u(u=3 x)\\$$
$$=\dfrac{3}{4} \dfrac{\pi}{2}-\dfrac{1}{4} \dfrac{\pi}{2}=\dfrac{\pi}{4}$$
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Single Correct Medium Published on 17th 09, 2020
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