Tricky question involving exact differentials

I've been asking around on this question for a bit, and though I have an answer for the first half of it, I'm a little bit iffy on the solidarity of the answer I have. Primarily, however, I could use a hand with the second half of this, since it isn't worded that well.

Show that $\displaystyle F=\left(\frac{x}{x^2+y^2}, \frac{y}{x^2+y^2}\right)$ and $\displaystyle G=\left(\frac{y}{x^2+y^2}, \frac{x}{x^2+y^2}\right)$ are such that one of $\displaystyle F.dP$, $\displaystyle G.dP$ is exact but the other is not $\displaystyle (dP=(dx,dy))$.

Let $\displaystyle C$ be the shorter arc of $\displaystyle x^2+y^1=1$ from $\displaystyle (1/\sqrt{2}, -1/\sqrt{2})$ to $\displaystyle (1/\sqrt{2}, 1/\sqrt{2})$.

Evaluate $\displaystyle \int_C F.dP$ and $\displaystyle \int_C G.dP$.

As you can see, this one is actually quite tough. If you could provide any assistance on it.