Compute the following integrals:
$\int_{\gamma_1}z^idz$, where $\gamma_1(t)=e^{it}, \frac{-\pi}{2}\leq t \leq \frac{\pi}{2}$.
$\int_{\gamma_2}z^idz$, where $\gamma_2(t)=e^{it}, \frac{\pi}{2}\leq t \leq \frac{3\pi}{2}$.
I think 1. is much easier than 2. since it does not cross the $Re(z)<0$. So for 1. $$\int_{\gamma_1}z^idz=\int^{\frac{\pi}{2}}_{\frac{-\pi}{2}}(e^{it})^i ie^{it}dt=i\int^{\frac{\pi}{2}}_{\frac{-\pi}{2}}e^{it-t}dt=i\frac{1}{i-1}e^{it-t}|^{\frac{\pi}{2}}_{\frac{-\pi}{2}}=\frac{1-i}{2}\left(e^{-\frac{\pi}{2}+i\frac{\pi}{2}}-e^{\frac{\pi}{2}-i\frac{\pi}{2}}\right)$$ However, for 2. we cannot apply the same method directly... Any help? Thanks!