From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$
Something I've never understood, though, is how this carries over to the complex plane. In that context, you basically have to replace line integrals with contour integrals. But doesn't that mean that we can go around the poles at $\{i,-i\}$ willy-nilly, drop some residue theorem into the mix, and end up with all sorts of different values for this integral?
I thought about this for awhile, and eventually decided that maybe this is what universal covering spaces are used for. Let $U$ denote the universal covering space of $\mathbb{C} \setminus \{i,-i\}$. Then if we can just find a way to "choose" two points $a,b \in U$ such that $a$ "is" $-1$ and $b$ "is" $+1$, then since there's only one path from $a$ to $b$ in $U$, maybe its possible to make sense of the integral $$\int_{-1}^1 \frac{1}{1+x^2}dx.$$
Question. Is this right? If so, I'd appreciate some more details.
If not, I'd appreciate a thoughtful discussion about the 'correct' way of thinking about this.