How to solve $\int_{-\infty}^{\infty}\frac{\sin^3 x}{x^3}dx$?

Question

The hint is to use integration by parts, but I really don't see how to integrate by parts here.

$$\int_{-\infty}^{\infty}\frac{\sin^3 x}{x^3}dx$$

This is a question from a complex analysis class, so I thought I would use something like semicircular contour.

Any help will be appreciated. Thanks!

https://math.stackexchange.com/questions/34436/evaluating-the-contour-integral-int-0-infty-frac-sin3xx3-math?rq=1 — Cheerful Parsnip, May 12 '17 at 03:24
Thanks for the link... that helped. My professor wants me to use integration by parts though. Any idea how to do it this way? — pamplemousse, May 12 '17 at 03:29
The linked question includes an answer using integration by parts. — Semiclassical, May 12 '17 at 05:03

score 0 · Answer 1 · answered May 12 '17 at 04:37

0

Hint:

Note that $\sin^3x=\frac{1}{4}(3\sin x-\sin 3x)$. So we only have to find $\int \frac{\sin x} {x^3}dx$. This can be done by using integration by parts repeatedly, first convert it to an integral with integrand $\frac{\cos x} {x^2} $

answered May 12 '17 at 04:37

CY Aries

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How to solve $\int_{-\infty}^{\infty}\frac{\sin^3 x}{x^3}dx$?

1 Answers1