How about splitting , into a much simpler component and a component that happens to be a conservative field like this

, where .

Since the curve is closed, the conservative part can be dropped from the line integral: you only need to integrate along that curve.

Are you asking what happens if you reverse the direction of the entire curve? - Well, in that case the line integral changes its sign.and how do i solve it when the arrow on the left hand side of the diagram changes to the other direction?