What steps do we usually perform to calculate integrals using Cauchy's theorem?

Question

I have to use only the theorem, I have the integral:

$$\int_C \frac{dz}{z^2+9}$$

if point $3i$ is inside the countor $C$ and $-3i$ is outside the countor $C$

I have Cauchy's formula on hand but do not know how to approach it, as far as I understand we do not care about points that are outside the countor, so we should not care of $-3i$

Answer 1

Cauchy's Formula:$$\frac{n!}{2\pi i}\int_C \frac{f(z)}{(z-a)^{n+1}}dz=f^n(a)$$ Now rewrite $$\frac{1}{z^2+9}=\frac{(z+3i)^{-1}}{z-3i}$$ We see now that $n=0$, $a=3i$, and $f(z)=\frac{1}{z+3i}$. Now just plug into the formula.