I have a non-compact smooth manifold $M$, a compactly supported smooth vector field $X$ on $M$ and a compactly supported smooth function $f$ on $M$.
Does there exist a compactly supported smooth function $F$ on $M$ such that $(dF)(X) = f$?
I thought about integrating $f$ along the integral curves of $X$ to get $F$. Does this actually always work or do I miss something (some nasty special cases which might occur)?