the domain and its derivative of $f(z)=\ln(z^2+2)$, where $\ln$ denotes the main branch of the log.

Question

Could you help me clarify this question? Thanks in advance

Be $f(z)=\ln(z^2+2)$, where $\ln$ denotes the main branch of the log.

1. Find the domain of $f$ and its derivative in this domain

The solution given is $\Bbb{C} - \{ iy: |y| \geq \sqrt{2}\}$

2. Computer $ \int _{\alpha} \frac{z}{z^2+2}\ dz $, where $\alpha$ is the differentiable curve with initial point at 0 and end point at $-1+i$

I know that the answer is $\frac{\ln2}{2} - \frac{i \pi}{4} $