The question is in trying to understand the "Argument Principle" Gamelin's Complex Analysis textbook (pages 224-225).
Suppose $f(z)$ is analytic on a domain $D$. For a curve $\gamma$ in $D$ such that $f(z) \neq 0$ on $\gamma$, we refer to \begin{equation} \frac{1}{2 \pi i} \int_{\gamma} \frac{f'(z)}{f(z)} \, dz = \frac{1}{2 \pi i} \int_{\gamma} d (\log f(z)) \end{equation} as the logarithmic integral of $f(z)$ along $\gamma$. Thus the logarithmic integral measures the change of $\log f(z)$ along the curve $\gamma$.
I couldn't understand what the author meant by "exact" and "closed but not exact". Any help in understanding these two concepts are much appreciated.
In this context, a differential $1$-form $\omega$ on a domain $D$ is:
Any closed differential form is exact, but the converse is not true due to topological obstructions.
Here is what the author means. The function $z\in D \mapsto \log |f(z)|$ is well defined and smooth (presumably, $f$ is asked to be non-vanishing earlier). Hence, the differential $1$-form $d\log |f|$ is exact by definition.
However, the map $z \in D \mapsto \arg f(z)$ is not really well defined: it is defined only up to a constant multiple of $2\pi$. Nevermind: the differential $1$-form $\omega = d \arg f$ is well defined (because two smooth locally defined determinations of $\arg f$ only differ by a constant). The notation $d \arg f$ is misleading since it suggests that $\arg f$ is well-defined and thus that $\omega$ is exact, but really, it isn't. What is true however is that it satisfies $d\omega = 0$, i.e, it is closed. One can prove this by noticing that $\omega$ is locally exact (meaning that locally, one can write $\omega = dg$ for some $g$).