TITLE: Evaluate $\int_0^\infty\frac {\sin^4(x)} {x^4} \operatorname dx$
QUESTION [9 upvotes]: How to evaluate the definite integral:
$$
\int \limits_0^\infty\frac {\sin^4(x)} {x^4} \operatorname dx
$$
Also provide the reference to various theorems to be to used to evaluate it!
Thank you.

REPLY [2 votes]: We can use a version of Parseval's equality:
$$\int_{-\infty}^{\infty} dx \, f(x)^2 = \frac1{2 \pi} \int_{-\infty}^{\infty} dk \, F(k)^2$$
where
$$F(k) = \int_{-\infty}^{\infty} dx \, f(x) \, e^{i k x}$$
Now, we may use the well-known FT:
$$f(x) = \frac{\sin^2{x}}{x^2} \implies F(k) = \pi \left (1-\frac{|k|}{2} \right ) $$
(This result may be easily derived using, e.g., the convolution theorem on the even more basic FT of $\sin{x}/x$.)
Thus the integral is $1/2$ of
$$\frac1{2 \pi} \pi^2 \int_{-2}^2 dk \, \left (1-\frac{|k|}{2} \right )^2  = \pi \int_0^2 \left ( 1-k+\frac14 k^2\right ) = \frac{2 \pi}{3}$$
or $\pi/3$.