I have been stuck on the following integral for some time:
$$ I = \int^{2 \pi}_0 \mathrm{d}\theta \frac{\cos \theta\left(x + \Delta\cos\theta\right)}{\sqrt{k + \left(x + \Delta\cos\theta\right)^2}} \exp\left[-\frac{\left(x + \Delta\cos\theta\right)^2}{\alpha^2}\right], $$ where $\Delta, k, \alpha \ll 1$ are small parameters. $x$ is another (small) coordinate that is later integrated over.
I have tried a few approaches:
Substitution: $y = x + \Delta \cos \theta$ is the obvious one, but I couldn't find a good way to handle the introduction of $\sin \theta$.
Using the fact that $\Delta$ is small parameter to drop the $\cos \theta$ terms. I don't like this because you basically drop all of the terms dependent on $\cos \theta$.
Mathematica: no luck...
I tried something along the lines of Laplace's method, but this came out to be zero, so I think I went wrong.
I'm not sure how to tackle this as a contour integral, I tried and it was a mess.
Do you have any hints that might help?
EDIT: Would the integral be easier to approach if: i.e. if: $$ I = \int^{2 \pi}_0 \mathrm{d}\theta \frac{\cos p \theta\left(x + \Delta\cos\theta\right)}{\sqrt{k + \left(x + \Delta\cos\theta\right)^2}} \exp\left[-\frac{\left(x + \Delta\cos\theta\right)^2}{\alpha^2}\right], $$