Solution
= Solution
Direct evaluation of the <integral> gives
$$
\begin{aligned}
\widetilde f(k)
&=\int_0^1e^{-ikx}\,dx-\int_{-1}^0e^{-ikx}\,dx\\
&=\frac{2-e^{ik}-e^{-ik}}{ik}
=-\frac{2i(1-\cos k)}{k}.
\end{aligned}
$$
The continuous value at $k=0$ is zero, in agreement with the vanishing <integral> of the <odd function> $f$. Thus
$$
\boxed{\widetilde f(k)=-\frac{2i(1-\cos k)}k}.
$$