求
$$
f(t)=\left\{\begin{array}{ll}
1 & |t| \leq 1 \\
0 & |t|>1
\end{array}\right.
$$
的傅立叶变换，并由此证明：
$$
\begin{aligned}
&\int_0^{+\infty} \frac{\sin \omega \cos \omega t}{\omega} d \omega=\left\{\begin{array}{cc}
\pi / 2 & |t| < 1 \\
\pi / 4 & |t|=1 \\
0 & |t|>1
\end{array}\right.
\end{aligned}
$$