I'm trying to compute $$\int_{\gamma}\frac{1}{z}\, dz $$ where $\gamma(t)=e^{-it}, t\in \,[0,8\pi].$
I've done this the following way: $$\int_0^{8\pi}f(\gamma(t))\gamma'(t)\, dt=\int_{0}^{8\pi}e^{it}\cdot-ie^{-it}\, dt=\int_0^{8\pi}-i \, dt=-8\pi i$$
I was happy with this but then I tried doing the calculation using the Fundamental Theorem of Calculus and found the answer to be $0$. Which one is correct? And why is there a difference?