My question states:
A function $h(x)$ has its Fourier transform given by
$$\mathcal{F}[sin(\alpha x)]\mathcal{F}[e^{-|x|}]$$
Show that $h(x)$ is of the form $A\sin({\alpha x})$ and find the constant $A$.
Hence, without directly computing the integral, show that
$$\int_{-\infty}^{\infty} \sin(\beta p) e^{-|x-p|}dp=\frac{2sin(\beta x)}{1+\beta^{2}}$$
I can calculate the integrals and derive the equation
$$\mathcal{F}[h(x)]=i \sqrt{\frac{\pi}{2}} (\delta(\alpha -k)-\delta (\alpha + k)) \sqrt{\frac{2}{\pi}} \frac{1}{1+k^{2}} = \frac{i}{1+k^{2}} (\delta(\alpha -k)-\delta (\alpha + k))$$
but I can't move from this. I am unsure how to calculate the inverse Fourier transform from this, and also I don't know why in the equation given in the question, there is constant $\beta$ in the denominator, while I have similar equation with variable $k$.