= Solution
Let $r(z)$ be the indicator of $[-1,1]$. The finite integral is exactly the <convolution> $p=q*r$. Its additional transform factor is
$$
\widetilde r(k)=\int_{-1}^1e^{-ikz}\,dz=\frac{2\sin k}{k},\qquad\widetilde r(0)=2.
$$
Applying the <convolution theorem> and part (d),
$$
\boxed{\widetilde p(k)=\frac{4\sin k(1-\cos k)}{k^3},\qquad\widetilde p(0)=2.}
$$
Equivalently this is $2(\sin k/k)[\sin(k/2)/(k/2)]^2$. The value at zero also follows from $\int p=(\int q)(\int r)=1\times2$.
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