Integrating $\int _{\gamma}\frac{dz}{z^2-1}$

Question

$$\int _{\gamma}\frac{dz}{z^2-1}$$

$\gamma:$ $it$ and $-1\leq t\leq 1$

The point $-1$ and $1$ are not on $\gamma$, can we say that the integral is $0$?

Answer 1

We can say that it is zero if $\gamma$ is a closed curve such that $-1$ and $1$ are not on $\gamma$ and inside the domain encircled by $\gamma$. But here $\gamma$ is not a closed curve., it is a segment.

On the other hand, we can evaluate the integral directly: after letting $z=it$, we get $$\int _{\gamma}\frac{dz}{z^2-1}=\int _{-1}^1\frac{idt}{(it)^2-1}.$$ Can you take it from here?