Usually, when one asks for the integral of $1/x$, they’ll be answered with “$ln(x)$”, but it’s “undefined” for negative values.
Sometimes, they’ll be answered with “$ln(|x|)$”, which is defined in the entire real numbers line (except $0$, but that doesn’t matter).
However, this only works for real values and I’ve never heard of complex values being discussed.
So that leaves me with the question of what that ever elusive integral is.
My first idea was that, on the unit circle, $1/x=x^*$, and that $x^*= e^{-i\theta}$. Integrating that gives $e^{i\theta}/-i\theta=ie^{-i\theta}/\theta$
I was going to check it on $1$, but I realized that division by $\theta$ doesn’t lend itself well to real values.
Other than that, I don’t really know what to do.