Once you’ve established that curl F = 0, you know that F is the gradient of some potential ⱷ. You can use the Fundamental Theorem to evaluate ∫F.dr — if only you could find the potential function. In this video I go over a straighforward method of finding the potential function using partial integration. (Note: the sound quality gets wonky near the end but you can still hear what I’m explaining perfectly well. Apologies and I’ll figure out eventually why that happens.)