Solution
= Solution
For $f=\mathbf1_{[-1,1]}$, part (b) gives $\widehat f(u)=2\sin u/u$. The <Plancherel theorem> in the stated Fourier convention gives
$$
\int_{\mathbb R}|f(x)|^2\,dx
=\frac1{2\pi}\int_{\mathbb R}|\widehat f(u)|^2\,du.
$$
Since the left side is two,
$$
2=\frac1{2\pi}\int_{\mathbb R}4\left(\frac{\sin u}{u}\right)^2du.
$$
The integrand is even, so the <sinc-squared integral> is
$$
\boxed{\ \int_0^\infty\left(\frac{\sin x}{x}\right)^2dx=\frac\pi2.\ }
$$