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
Questions 203525
Subjects 9
Chapters 126
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