I personally think until you understand the geometry of a multifunction (it's real and imaginary Riemann surfaces), contour integrations over these surfaces will always be difficult to understand. Take time to draw them and understand them, then the integrals become a breeze.
I want to add something to what I said. The function has a primitive on . On this "slit plane" (or cut-plane) . However, on the primitive is not even continous (because the logarithm jumps its value as it comes around). Therefore, if you are trying to compute the line integral from to and the curve that you are integration does not cross over then the integral is simply . However, if the curve does cross over then you need to be careful and cannot use this fundamental theorem for line integrals.