I have to apply the residue theorem on the linked complex integral but I am confused as to which poles are inside the proposed closed path and which are not. I would appreciate the help.
(I don't have enough points to post a picture so here is the link.)
The winding numbers of $C$ with respect to $-1$, $0$, and $1$ are $1$, $2$, and $-1$ respectively. Therefore\begin{multline}\int_C\frac{e^z}{z^2(1-z^2)}\,\mathrm dz=\\=\operatorname{res}_{z=-1}\frac{e^z}{z^2(1-z^2)}\,\mathrm dz+2\operatorname{res}_{z=0}\frac{e^z}{z^2(1-z^2)}\,\mathrm dz-\operatorname{res}_{z=1}\frac{e^z}{z^2(1-z^2)}\,\mathrm dz.\end{multline}