TITLE: Integral using Parseval's Theorem
QUESTION [5 upvotes]: How would I integrate 
$$\int_{-\infty}^{+\infty} \frac{\sin^{2}(x)}{x^{2}}\,dx$$  using Fourier Transform methods, i.e. using Parseval's Theorem ? 
How would I then use that to calculate: 
$$\int_{-\infty}^{+\infty} \frac{\sin^{4}(x)}{x^{4}}\,dx$$?

REPLY [3 votes]: Both integrals can be computed through integration by parts, since:
$$ \int_{\mathbb{R}}\frac{\sin^2 x}{x^2}\,dx = \int_{\mathbb{R}}\frac{\sin(2x)}{x}\,dx = \pi$$
and: 
$$ \int_{\mathbb{R}}\frac{\sin^4 x}{x^4}\,dx = \int_{\mathbb{R}}\frac{\sin(2x)-\frac{1}{2}\sin(4x)}{3x^3}\,dx=\int_{\mathbb{R}}\frac{\cos(2x)-\cos(4x)}{3x^2}\,dx=\frac{2\pi}{3}.$$
The general integral
$$\int_{\mathbb{R}}\left(\frac{\sin x}{x}\right)^n\,dx $$
can be computed from the pdf of the Irwin-Hall distribution.