Let C be any simple closed contour inside the annulus 4 < |z| < 6. Show that there holds:
$$ \int_C \frac{dz}{z^2+1} = 0$$
To begin: I know that there are poles at $\pm i$ and that the integral can be rewritten as
$$ \int_C \frac{dz}{(z+i)(z-i)}$$
I think there is a theorem on multiply connected domains that I need to understand, but am unsure as to how one would approach this question.
Either both poles or neither are inside the contour. The residues at the poles are .... Apply the Residue Theorem.