TITLE: Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.
QUESTION [5 upvotes]: I am trying to compute the following integral using the Residue Theorem but am quite stuck:
$$\int_0^\infty \frac{\sin^2x}{x^2}dx$$
I have tried applying Jordan's lemma, having written $\sin(x)$ as $\dfrac{\mathrm{e}^{ix}-\mathrm{e}^{-ix}}{2i}$, to not much avail. I have also tried using a rectangular integration path but it didn't get me far. I'd be grateful for an insightful advice.

REPLY [2 votes]: In this answer, it is shown using contour integration and the binomial theorem that
$$
\int_0^\infty\left(\frac{\sin(z)}{z}\right)^n\mathrm{d}z
=\frac{\pi}{2^n(n-1)!}\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\binom{n}{k}(n-2k)^{n-1}
$$
Plugging in $n=2$ gives $\dfrac\pi2$.