Let $D$ be the annulus $6\lt |z| \lt 8$ and let $C$ be any simple closed contour inside $D$. Show that there holds: $$\int_C \frac{dz}{z^2+1} =0$$
This has two singular points, $z=\pm i$, these are both outside of the annulus and hence by Cauchy Goursat theorem, we know that when $C$ is a simple closed contour in $\Bbb C$ and where this is analytic in and on any contour within $D$, and hence the integral on any contour in $D$ evaluates to $0$
Is that all I need?