Find a parametric equation to solve a line integral Find 
$ \int_C \frac{dz}{z} $ from $1 - 5i$ to $5 + 6i$
I want to use 
$$\int_C f(Z) dz = \int_a^b f[z(t)]z'(t)dt$$
My guess is that I need to find $z(t)$ such that 
$$z(a) = 1 - 5i$$
$$z(b) = 5 + 6i$$
But how?

You can actually use any path from $a$ to $b$.  You should get the same answer as long as the path does not include the origin.  This is the Principle of Path Independence.  It follows from the Cauchy-Goursat Theorem.
