Integrate from 0 to infinity
$$\int\limits_{0}^{\infty}\left(\frac{\sin(ax)}{x}\right)^2 dx , a \neq 0 $$
I tried evaluating the indefinite integral of that function using Sine integral. But I failed to do it. I am having no idea to evaluate the definite integral. I seek for help
$\endgroup$ 52 Answers
$\begingroup$First from here it is easy to prove using the changes $u=2y$ that $$\int_0^\infty \frac{\sin(u)}{u}dx= \int_0^\infty\left(\frac{\sin(u)}{u}\right)^2dx$$
Inserting the change of variables $u=ax$ one gets
$$\int_0^\infty\left(\frac{\sin(ax)}{x}\right)^2dx = a\int_0^\infty\left(\frac{\sin(ax)}{ax}\right)^2d(ax)\\= a \operatorname{sign}(a)\int_0^\infty\left(\frac{\sin(u)}{u}\right)^2dx =|a|\int_0^\infty \frac{\sin(u)}{u}dx =\color{blue}{\frac{|a|π}{2} }$$
this comes from the Dirichlet integral Evaluating the integral $\int_0^\infty \frac{\sin x} x \ dx = \frac \pi 2$?
$\endgroup$ 6 $\begingroup$Hint
$$\int\frac{\sin^2(ax)}{x^2} d x = - \frac{\sin^2(a x)}{x} + \int \frac{2a \sin(a x)\cos(a x)}{x} d x$$
and $2 a\sin(a x)\cos(a x) = a \sin(2 a x)$
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