to evaluate $\int_{-\infty}^{\infty} (\frac{\sin x}{x})^{2}\ dx$

Question

I have to evaluate $$\int_{-\infty}^{\infty} \bigg(\frac{\sin x}{x}\bigg)^{2}\ dx$$ and hint says that use Plancherel theorem. Now, in my notes Plancherel theorem is just a statement that we can extend fourier transform map which was defined originally from $S(\mathbb R)$ to $S(\mathbb R)$ to a map from $L^2(R)$ to $L^2(R)$. But I am not getting at all how Plancherel theorem is going to help me evaluate this integral.Any help.Thanks.

Answer 1

Hint What is the Fourier transform of the rectangular function?

Answer 2

Apply integration-by-parts,

$$\int_{-\infty}^{\infty} \bigg(\frac{\sin x}{x}\bigg)^{2}\ dx = -2\int_{0}^{\infty} {\sin^2x}\ d(\frac1x) =2\int_{0}^{\infty}\frac{\sin 2x}{x}dx$$ $$=2\int_{0}^{\infty}\frac{\sin t}{t}dt=2\cdot \frac\pi2=\pi$$