Evaluating a real integral using complex methods

Let's say I want to evaluate the following integral using complex methods -

$$\displaystyle\int_0^{2\pi} \frac {1}{1+\cos\theta}d\theta$$

So I assume this is not very hard to be solved using real analysis methods, but let's transform the problem for the real plane to the complex plane, and instead calculate -

$$\begin{aligned}\displaystyle\int_0^{2\pi} \frac {1}{1+\cos\theta}d\theta \quad&\Longrightarrow \quad [ z=e^{i\theta} , |z| =1]\\
&\Longrightarrow \quad\displaystyle\int_{|z|=1} \frac {1}{1+\frac{z+\frac{1}{z}}{2}}\frac{dz}{iz}\end{aligned}$$

So now after few algebric fixed this is very easily solvable using the residue theorem.

My question is why can I just decide that I want to change the integration bounds for $[0,2\pi]$ to $|z|=1$. If I wanted to change the integrating variable to $z=e^{i\theta}$ aren't the integration bounds suppose to transform to $[1,1]$ (because $e^{i2\pi k}=1$)? I'm just having hard time figuring out why is this mathematicaly a right transform.

Thanks in advance!