How do we find
$$\int_{-\infty}^{\infty}\dfrac{\sin(ax)}{\sqrt{(b-x)^2+c^2}}dx$$
I have no idea on how to even begin approaching this. Can I get a hint?
EDIT:
I've got to this integral from a physics exercise in electromagnetism, not sure how well it fits here, but here is the exercise:
In an area $y<0$ there's emptiness, and in an area $y>0$ there's magnetism: $\overrightarrow{M}=M_0cos(\beta x)e^{-ay}\hat{y}$. Find the magnetic potential in every point in the space.
I used this formula: $\overrightarrow{J}_M=\overrightarrow{\nabla} \times \overrightarrow{M}$ and substituted it in $\overrightarrow{A}(\overrightarrow{R})=\frac{\mu_0}{4\pi}\int\int\int_{v'} \frac{\overrightarrow{j}(\overrightarrow{R'})}{|\overrightarrow{R}-\overrightarrow{R'}|}dV'$ and ended up with the integral in the question.
Assume $a,c>0$. Put $x=b-t$ to get $$\sin(ab)\int_{-\infty}^\infty\frac{\cos(at)\,dt}{\sqrt{c^2+t^2}}-\cos(ab)\int_{-\infty}^\infty\frac{\sin(at)\,dt}{\sqrt{c^2+t^2}}.$$ The second integral is zero, and the first one is $2K_0(ac)$ (see Basset's integral).