Let $\gamma:[0,1] \to \mathbb{C}$ be the curve $\gamma(t) = e^{2\pi i t}, 0 \le t \le 1$. Find, giving justifications, the value of the contour integral $$ \int_\gamma \frac{dz}{4z^2-1}. $$
I know the Cauchy residue theorem and how to apply it. But I couldn't the let part in the question.
The curve described parametrically by $\gamma(t)=e^{i2\pi t}$, $0\le t\le 1$ is simply the unit circle centered at $z=0$ (i.e., $|z|=1$).
Hence, we have
$$\begin{align} \int_\gamma \frac{1}{4z^2-1}\,dz&= \oint_{|z|=1}\frac{1}{4z^2-1}\,dz\\\\ &=2\pi i \text{Res}\left(\frac{1}{4z^2-1}, z=\pm 1/2\right)\\\\ &=2\pi i \left(\frac14-\frac14 \right)\\\\ &=0 \end{align}$$