Prove that ${x dy - y d x\over x^2 + y^2}$ is not exact
By Green's theorem,
$$\int_{S^1} \psi = \int_{S^1} x\, dy - y\, dx = \iint_{D^2} \text{div}(\langle x,y\rangle)\, dA = 2\cdot \text{Area}(D^2) = 2\pi$$
Alternatively, parametrize $S^1$ by setting $x = \cos(t)$, $y = \sin(t)$, $0 \le t \le 2\pi$. Then $$\int_{S^1} \psi = \int_0^{2\pi} (\cos(t)\cdot \cos(t) - \sin(t)\cdot (-\sin(t)))\, dt = \int_0^{2\pi} (\cos^2(t) + \sin^2(t))\, dt = 2\pi$$
Either way, $\int_{S^1} \psi \neq 0$.