The problem is: Use the definition of complex integral $\int_Cf(z)d|z|$ over a curve to evaluate the integral $\int_C\frac{d|z|}{\overline{z}}$, where $C=\{e^{it}:t\in[0,2\pi]\}$.
The $d|z|$ part is confusing me. The question seems pretty easy. I am sure I can easily solve it if I knew the definition of $\int_Cf(z)d|z|$. I hope someone can help me with this. Thanks!
I think it is a typo, and should be printed as $|dz|$. For your curve $C$ this would lead to $|dz|=dt$ $(0\leq t\leq 2\pi)$.
Note that $d|z|$ means the infinitesimal change in $|z|$, hence would be $\equiv0$ along $C$.