How to deal with $\int\limits_{L}z\cos(2\pi z \bar{z})\,dz$?

Question

The original integral is:

$$\int\limits_{L}z\cos(2\pi z \bar{z})\,dz$$

where $$L = \{|z| = 1; \quad 0 \le \arg \le \pi \}$$

very well seen that integration region is the semi-circle $[0;\pi]$ but I am totally confused how to deal with cosine's arg :(

Answer 1

Notice that because $z\bar z= |z|^2=1$, the integral becomes, $$\int \limits_{L} z \cos 2\pi \, dz = \int \limits_{L} z \, dz = \,?$$

Answer 2

$\int\limits_{L}z\cos(2\pi z \bar{z})\,dz= \int_0^{\pi}e^{it} \cos (2 \pi e^{it}e^{-it}) i e^{it} dt=i \int_0^{\pi}e^{2it} dt=0$.