Finding the complex integral along an arc

Question

How can we evaluate complex expressions like these
$\int_C(Z-Z^2)dZ$ where $C$ is the upper half of the circle $|Z-2|=3$

Answer 1

For that particular $C$, write $Z=2+3 e^{i t}$, $dZ=i 3 e^{i t} dt$, where $t \in [0,\pi]$. The integral may now be written as a simple, definite integral:

$$-i 3 \int_0^{\pi} dt \, e^{i t} (2+3 e^{i t}) (1+3 e^{i t}) = -i 3 \int_0^{\pi} dt (2 e^{i t} + 9 e^{i 2 t} + 9 e^{i 3 t}) = 30$$