Integral of $1/z$ over $\partial D(0,1)$

Question

I've seen in my book that $$\int_{\partial D(0,1)} \frac{dz}{z}=1$$.

When I try to prove it, I have:

$$\int_{\partial D(0,1)}\frac{1/z}=\int_{0}^{2\pi}\frac{dt/e^{it}}dt=0$$

Could you please help me see my error? Thank you!

Answer 1

Hint. Note that by letting $z=e^{it}$ then $dz=e^{it}i\,dt$, therefore $$\int_{\partial D(0,1)} \frac{dz}{z}=\int_{t=0}^{2\pi} \frac{e^{it}i\,dt}{e^{it}}.$$ Can you take it from here?

P.S. You should obtain $2\pi i$ and not $1$.