= Exponential kernel convolved with an interval indicator
{title2=$h(x)=\int_{-b}^be^{-a|x-y|}\,dy$}
For $a,b>0$, the convolution of $e^{-a|x|}$ with the indicator of $[-b,b]$ is $2e^{-a|x|}\sinh(ab)/a$ outside the interval and $2[1-e^{-ab}\cosh(ax)]/a$ inside it. With Fourier convention $\widetilde f(k)=\int f(x)e^{-ikx}\,dx$, their transforms are $2a/(a^2+k^2)$ and $2\sin(bk)/k$. Fourier inversion therefore evaluates $\int_{\mathbb R}\sin(bk)e^{ikx}/[k(a^2+k^2)]\,dk$ as $\pi h(x)/(2a)$.
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