Line integral equals zero. Currently going through some line integral problems and in the worked example provided, the line integral evaluated to become zero. The curve C that they have provided was a simple closed curve (it was a half-circle). Does the answer being zero have anything at all to do with the fact that C is closed and simple? Or was it merely a coincidence that it turned out to be zero?

Yes if $C$ is a simple closed curve, $\textbf{v}:\mathbb{R}^m\rightarrow \mathbb{R}^m$ and$$\oint_{C} \textbf{v}\;d\textbf{r}=0,$$ then it follows that $\textbf{v}=\
abla f$ for some scalar function $f:\mathbb{R}^m\rightarrow \mathbb{R}$.