I'm looking for a way of evaluating $$\int_0^\pi\sin x \exp(-a/\sin x)dx$$ to get a second order Bickley function $K_2(a)$, which is basically the same integral, but $\cos x$ instead of $\sin x$ and the limits change from $0$ to $\pi/2$, which is understandable.
I'm a bit lost what kind of variable substitution could I do. Any suggestions? I have tried $-a/\sin x = u$, but that doesnt seem to give reasonable results. Thanks in advance!