Question: Directly compute the following line integral:
$$\int _\gamma \frac{dz}{z+4}$$
where $\gamma$ is the circle of radius 1 centered at -4, oriented counterclockwise.
Answer: Let $\gamma (t) = -4 + e^{it}$, where $t \in [0, 2\pi]$.
Then, $\gamma ' (t) = ie ^{it}$.
Then, we compute the line integral:
$$\int _\gamma \frac{dz}{z+4} = \int _0^{2\pi}\frac{ie ^{it} dt}{-4 + e^{it} + 4}$$
$$ = \int _0^{2\pi}\frac{ie ^{it}}{ e^{it}}dt$$
$$ = \int _0^{2\pi}i dt$$
$$ = it \Big|_{t = 0}^{t = 2\pi}$$
$$ = 2\pi i$$
