Residue theorem sounds fine, if you're allowed to use it.
I need to integrate f(z)dz, where f(z) = (z+i)/[(z^3)+(2z^2)], along C, where C is the circles |z| = 1, |z+2-i| = 2, and |z-2i| = 1, each traversed once counterclockwise (so I'll end up doing the integral once for each C). I started to evaluate it along the first circle using the usual method for line integrals (parameterizing C and integrating f(C(t))C'(t)dt from 0 to 2pi), but I end up with an messy integral. Should I be using a different approach? I'm just now reading up on the residue theorem, should I use that instead?
Sorry if my post is hard to read, I'm planning on learning LaTeX in the near future.