Evaluate $\int_{\mathbb{R}} \frac{\sin^{4}(x)}{x^2}dx$

Question

Need help evaluating

$$\int_{\mathbb{R}} \frac{\sin^{4}(x)}{x^2}dx$$

I know this is equivalent to $\int_{\mathbb{R}} \text{sinc}^{2}(x)\sin^2(x)dx$, but I'm not sure if this brings me any closer.

Any help?

Answer 1

If we use integration by parts and compute a Fourier sine series we have:

$$ \int_{0}^{+\infty}\frac{\sin^4(x)}{x^2}\,dx = \int_{0}^{+\infty}\frac{4\cos(x)\sin^3(x)}{x}\,dx = \int_{0}^{+\infty}\frac{\sin(2x)-\frac{1}{2}\sin(4x)}{x}=\color{red}{\frac{\pi}{4}}$$ since for every $n\in\mathbb{N}^+$, $$ \int_{0}^{+\infty}\frac{\sin(nx)}{n}\,dx = \int_{0}^{+\infty}\frac{\sin z}{z}\,dz = \frac{\pi}{2}.$$