Exponential kernel convolved with an interval indicator ($h(x)={\int }_{-b}^b{e}^{-a|x-y|}dy$)

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 $\tilde{f}(k)=\int f(x)e^{-ikx}dx$, 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 $\pi h(x)/(2a)$.