Evaluate the integrals $$\int_{\gamma}z^{n}dz$$ for all integers $n$. Here $\gamma$ is any circle centered at the origin with the positive (counterclockwise) orientation.
I don't know how to proceed. I couldn't find a "quick" way to do this, so I thought I'd use induction over $n$, but it seems like unnecessary work. Is there a clever way to calculate this integral?
My attempt. $z(t) = re^{it}$, so $$\int_{\gamma}z^{n}dz = \int_{0}^{2\pi}(re^{it})^{n}ire^{it}dt = ir^{n+1}\int_{0}^{2\pi}(e^{it})^{n+1}dt.$$
Your work looks perfectly good to me, you just need to finish it. If $n \neq -1$, then $n+1 \neq 0$, so $\int_0^{2 \pi} (e^{it})^{n+1} = 0$ (why?) If $n=-1$, then your reasoning shows the result is $ i \int_0^{2 \pi} 1 dz = 2 \pi i $.