When do I know I need to use Cauchy's Integral or Cauchy's Theorem?

Question

I do not fully understand the concept of each theory as they both look the same to me. When given an integral such as $$\int_\Gamma \frac{1}z dz$$

and a path $$\Gamma=\{ z:{\mid z-1 \mid=2}\} $$

how do I know which one to use to evaluate the integral?

Answer 1

Since the region includes the origin, and the function $\frac1z$ is not defined there, hence not holomorphic, Cauchy's theorem doesn't apply.

Cauchy's integral formula does though, and the integral equals $2\pi i$...