integral involving exponential with complex z

Question

Identify $\int _C e^z dz$ for any loop (including closed path).

My approach is since the integral of $e^z$ is $e^z$, I can use it to conclude that the answer is 0. But just not sure which corollary was used.

Answer 1

$\int_{\gamma} f'(z)dz=\int_a^{b} f'(\gamma (t)) \gamma'(t)dt =f(\gamma(b))-f(\gamma(a))$ for the path from $\gamma(a)$ to $\gamma(b)$ given by the piece-wise continuously differentiable function $\gamma: [a,b] \to \mathbb C$. In the present case $f(z)=f'(z)=e^{z}$.