Let $r$ be the function from $\mathbb{R}$ to $\mathbb{C}$ such that, for all real $x$,
$$r(x) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \mathrm{e}^{\mathrm{i}x\xi} \rho(\xi) \,\mathrm{d}\xi$$
where $\rho \in \mathcal{C}^\infty(\mathbb{R})$ is constant equal to 1 on $[-1,1]$ and constant equal to 0 on $\mathbb{R} \setminus [-2,2]$.
Show that $x \mapsto x^2 r(x)$ is bounded on $\mathbb{R}$ and deduce that $r$ is integrable and bounded on $\mathbb{R}$.