I want to evaluate the following integral $\int_{\gamma} 1/z \, dz$ where $\gamma$ is the semi-circle in the upper half plane from $1$ to $-1$.
Can I just use the Fundamental Theorem of Calculus here? I know that if $\gamma$ crosses the negative real axis I can't but what about here where it touches it?
Why not directly as line integral?
$$z=e^{it}\;,\;\;t\in [0,\pi]\implies dz=ie^{it}dt\implies$$
and now :
$$\int_\gamma\frac{dz}z=\int_0^\pi\frac{i\,e^{it}}{e^{it}}dt=\pi i$$
If you insist in using the FTC: take Log$\,z\;$, but remove the ray $\;iy\;,\;\;y\in(-\infty,0]\;$ , thus taking the argument of a number in the remaining domain to be an angle in $\;\left(-\frac\pi2,\,\frac{3\pi}2\right)\;$ , and thus:
$$\left.\int_\gamma\frac{dz}z=\text{Log}\,z\right|_1^{-1}=\text{Log}\,(-1)-\text{Log}\,1=\require{cancel}\cancel{\log1}+i\arg(-1)-(\cancel{\log1}-i\arg 1)=i\pi+i\cdot0$$