I have a proposition here with a suggest for a proof. I did not find this in my textbook, which surprised me, since it seems so obvious and simple, and seems to me to give a very easy way to know for sure whether a particular vector field is conservative or not (at least if it is easily integratable). The fact that it is not in my textbook makes me think I made a mistake.
Here it is:
Proposition. Let $F(x,y) = \binom{f(x,y)}{g(x,y)}$ be a vector field. If $$\int f(x,y)dx = \int g(x,y)dy=\phi(x,y),$$ for some $\phi$, then $F$ is a conservative vector field with $F = \nabla \phi$.
proof. Let $$\phi(x,y)= \int f(x,y)dx = \int g(x,y)dy$$ Then by the fundamental theorem of calculus, $\delta\phi/\delta x=f(x,y)$, and $\delta\phi/\delta y = g(x,y)$.
Hence, $F=\nabla \phi$, which completes the proof.
If this proposition is correct, then all we need to do to check whether a vector field is conservative is to check the integral of both dimensions of the field and see whether they are the same.