The first part of the question asks to define a branch of $\log(z+i)$ which is holomorphic in the cut-plane $\mathbb{C} \setminus \{z : \text{Re} (z) = 0, \text{Im}(z) ≤ -1\}$. I've defined this as $\log(r)+i\theta$ with $z + i = re^{i\theta}$, $r>0$ and $\theta \in (-\pi/2, 3\pi/2)$.
The next part of the questions asks: by integrating $\log(z+i)/(z^2+1)$ around a suitable closed path, evaluate $$ \int_{-\infty}^{\infty} \frac{\log(x+i)}{x^2+1} \text{d}x. $$
I think I'm meant to integrate along a semicircle of radius $R$, then subtract the integral along the arc of the semicircle and taking $R \to \infty$. Using the residue theorem, I've calculated the residue at $z=i$ to be $\log(2i)/2i$, so the integral along the semicircle is $\pi\log(2i)$. My question is, how am I meant to integrate along the arc? I suspect that it goes to 0, but I can't seem to find why.