Evaluate the integral $\int_{|z|=1} \frac{|dz|}{z}$

Question

Evaluate the integral $\int_{|z|=1} \frac{|dz|}{z}$

My idea:

Let $z=e^{i\theta}$

then $dz=ie^{i\theta}d\theta$

therefore $\int_{|z|=1} \frac{|dz|}{z}=\int ^{2\pi}_{0} e^{-i \theta}|d\theta|$

from here how to we processed

Answer 1

Since $\theta$ varies from $o$ to $2\pi$,so $|d\theta | = d\theta$,so the problem is to find $\int_{0}^{2\pi} e^{-i\theta} d\theta = \int_{0}^{2\pi} \cos(\theta) - i\sin(\theta) d\theta = (0-0) + i.(1-1) = 0$.

Also integrating wrt $|dz|$ is a bit new scenario BTW can we do this?