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 and are such that one of , is exact but the other is not .

Let be the shorter arc of from to .

Evaluate and .

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