Contour Integral around square

Question

Calculate the following around a square of side-length 4 centered on the origin, with sides parallel to the coordinate axes:

$$\quad \quad \oint \frac{\sin(z)dz}{z^2(1+z^2)}$$

I think I would have to apply Cauchy's Residue Theorem but I am not sure?

Answer 1

The residue theorem should work. You have simple poles at $0$ and $\pm i$.

Compute the residues, add them up, and multiply by $2\pi i$.

I get a residue of $1$ at $0$. And $\dfrac{-\sin i}{2i}$ at $\pm i$.

Thus we get $2\pi i(1-\dfrac {2\sin i}{2i})=2\pi i-2\pi\sin i$.