Can I get some guidance on solving $\int_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2} \, dx$? I am trying to evaluate:
$$I = \int_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2} \, dx.$$
Using a contour semi-circle (upper plane), I can get:
$$ \oint_{C} f(z) \,dz = \oint_{C} \frac{1 - e^{2iz}}{z^2} \, dz.$$
The whole issue is the $z^2$. I cannot use the residue theory, because it lies on the contour.
I don’t want a full solution. I really want to try on my own, I just need some guidance!
 A: The integrand contains a removable singularity at $z=0$. That is, the integrand is an entire function.

This answer shows how to compute
$$
\int_{-\infty}^\infty\left(\frac{\sin(x)}x\right)^n\mathrm{d}x
$$
by first showing that it equals
$$
\int_{-\infty-i}^{\infty-i}\left(\frac{\sin(x)}x\right)^n\mathrm{d}x
$$
Using $\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$ and the binomial theorem, we apply contour integration to get
$$
\int_{-\infty}^\infty\left(\frac{\sin(x)}x\right)^n\mathrm{d}x=\frac{2\pi}{2^n(n-1)!}\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\binom{n}{k}(n-2k)^{n-1}
$$
For $n=2$, this gives
$$
\int_0^\infty\frac{\sin^2(x)}{x^2}\mathrm{d}x=\frac\pi2
$$

Another approach, using Riemann Sums, is shown in this answer.
A: Avoid $z=0$ with a semicircle $c_\epsilon$ of radius $\epsilon>0$ small (clockwise sense). You will have to compute
$$
\lim_{\epsilon\to0}\int_{c_\epsilon}\frac{1-e^{2iz}}{z^2}\,dz.
$$
A: How about good old integration by parts?
$$ \int_{-\infty}^{+\infty} \frac{\sin^2(x)}{x^2} dx = \Big|_{-\infty}^{+\infty} -\frac{\sin^2(x)}{x} + \int_{-\infty}^{+\infty} \frac {2 \sin(x)\cos(x)}{x} dx = 0+ \int_{-\infty}^{+\infty} \frac{\sin(2x)}{x} dx = \int_{-\infty}^{+\infty} \frac{\sin(x)}{x} dx = \pi  $$ where at the end I use the substitution $u=2x$ and the final integral is well-known to equal $\pi$ ( if you are not familiar with this result, tell me, I can prove that too)
A: Another way to think about this is to note that $$\displaystyle \int\frac{\sin^2(x)}{x^2} dx = \int \frac{\sin(x)}{x} dx$$ Then you can use the well know solution to the Dirichlet integral. If you don't see how the two integrals are related I can prove it two. Hint: use integration by parts.
