Integral $\int_{|z-a|=r} \frac{dz}{z}$ How am I allowed to modify the integral
$$\int_{|z-a|=r} \frac{dz}{z}$$
?
It looks reminiscent of the Cauchy integral formula, but is it related?

If you don't have access to the residue theorem (or Cauchy's integral formula) just yet, but have seen Cacuchy's integral theorem, you can argue in the following way:
First, if $z=0$ is outside the disc $|z-a| < r$ (i.e. if $r < |a|$), the integrand is holomorphic on and inside the curve. Cauchy's integral theorem then shows that the integral is $0$.
If $r > |a|$, the problematic point $z=0$ is inside the curve, but Cauchy's integral theorem shows that the integral you are looking for is equal to 
$$
\int_{|z|=\varepsilon} \frac{dz}{z}
$$
for some suitably small $\varepsilon$. (The integrand is holomorphic on the region between the two circles.) This new integral is easy to compute via parametrization.
Finally, if $r = |a|$ you are in trouble, since the integrand has a (non-integrable) singularity on the curve.
