Disclaimer: I know the Residue Theorem could be used for this, but we cannot use it as we have not proven it in class.
In my complex analysis course, I am given a function $f(z)=\frac{1}{z^2-z}$ and a domain on which it is holomorphic $f(z)=\mathbb{C}\setminus\{-1,0,1 \}$. I am then tasked with `calculating' the difference
$$\int_{\gamma_1}f(z)dz-\int_{\gamma_0}f(z) dz $$
where $\gamma_1$ and $\gamma_0$ are arbitrary open curves indicated by a drawing in this domain (I will attach the drawing). In our class, we were presented the concept of a set being simply connected if any two curves in that domain are homotopy equivalent. From this definition, I am pretty sure $U$ is not simply connected, and thus $\int_{\gamma_1}f(z)dz-\int_{\gamma_0}f(z)dz\neq 0$. Based on the fact that we recently covered the complex exponential function and logarithm, I imagine that I am somehow to use that. But I am unsure how, given the fact that the curves are arbitrary, but at least vaguely circular. 
My one idea is that if this is meant to form a circle, then perhaps we could use the principal branch of the logarithm function on this domain and then parameterize the curves accordingly. But then I would wonder how this would differ from another subpart of the problem where the two curves go around both the points $0$ and $1$ - perhaps the radius of the parameterization would change? Any assistance would be appreciated.