What are all the possible values of the contour integral $\int_C \frac{dz}{z^2+1}$ where $C$ is piecewise smooth with starting point $0$ and ending point $1$
First, note that $C$ must be restricted to not pass through $\pm i$. From here, the method I used depended on arguing that the integral's value is path independent, so for simplicity I took the parametrization $z(t) = t$ for $t \in [0,1]$. Finding the possible values from this path would then give all of the possible values the integral can actualize.
The integral can be split up via partial fractions to: $$\frac{i}2(\int_0^1 \frac{dt}{t+i} - \int_0^1 \frac{dt}{t-i})$$
Which when evaluated will give that the possible values for $\int_C \frac{dz}{z^2+1}$ are $\frac{\pi}{4} \pm \pi k \space$ for $k \in \mathbb{N}$
Upon reading the answer in the back of my Complex Analysis textbook, my result agreed with theirs.
Despite getting the correct answer, I'm having some trouble with my argument's validity. Particularly, "that the integral's value is path independent"
Usually, the argument I use for path independence relies on Cauchy's Theorem: If we have 2 paths connecting $0$ to $1$, say $C$ and $C^*$, then we could write $$0 = \oint_{C \to -C^*} f(z)dz = \int_C f(z)dz + \int_{-C^*} f(z)dz =\int_C f(z)dz - \int_{C^*} f(z)dz \implies \int_C f(z)dz = \int_{C^*} f(z)dz$$
However, the first step above (as I understand) only holds if $f$ is analytic on a simply-connected domain, $E$. The smallest we can take $E$ to be is the union of $C \to -C^*$ with its interior. If this domain contains $i$ or $-i$, then $f$ is no longer analytic on $E$ (at those points), so I shouldn't be able to say that. Am I missing something?
In particular, am I wrong to assume path independence and happened to get the correct answer for the wrong reason, or is there another way to explain the path independence while avoiding analyticity of $f$ at these points?
The whole point is that the integral is path-dependent.
If you have two contours joining $0$ and $1$ (and avoiding $\pm i$), you can compose one with the reverse of the other to get a closed contour $C'$. The integral over this new contour is $2\pi i(mr+ns)$ where $m$, $n\in \Bbb Z$ are the winding numbers of $C'$ about $i$ and $-i$, and $r$ and $s$ are the residues at $i$ and $-i$. You can achieve any pair of integers for $m$ and $n$. So you need to calculated these residues, and then the general value for your integral is $\pi/4+2\pi i(mr+ns)$.