I have a problem as follows:

Let $\displaystyle \gamma=\beta+[e^2\pi,1]$ where $\displaystyle \beta$ is given by $\displaystyle \beta(t)=e^{t+it}$ for $\displaystyle 0\leq t\leq 2\pi$. Evaluate $\displaystyle \int_\gamma z^{-1} dz $.

I know that I need to parameterize the path and I have come up with the path as $\displaystyle [e^2\pi,1]$. I also know that I need to integrate this problem along the path and then integrate the function and add the parts together. However, I can't seem to come up with the right way to accomplish this task. I will appreciate your help. Thank you.