I have to find the Fourier Sine transform of $f(x)=1$ when $|x|<a$ and $f(x)=0$ when $|x|\ge a$ and hence show that $$\int_0^\infty {\sin(t)\over t} dt =\pi/2$$ and $$\int_0^\infty \left({\sin(t)\over t}\right)^2 dt =\pi/2$$
I found the first two parts. How shall I proceed in order to prove the last part? please help.
Hint: Use the fact that the Fourier transform is an $L^2$ isometry, i.e.
$$\langle \mathcal{F}f,\mathcal{F}f\rangle = \langle f,f\rangle,$$
where in your case $f(x) = \begin{cases} 1 & |x|\le 1 \\ 0 & |x|>1\end{cases}.$