How to find counter example to show that a vector field is not conservative?

Question

I know that to show it is not a conservation vector field I could find a closed parameterized path and then calculate the line integral such that it is not equal to zero, I've tried using a unit circle but I couldn't do this integration, what other paths could I try?

Answer 1

An easier way to do it is to just show that $\nabla \times F \neq 0$. This implies that $F$ is not conservative