I assume that by "where A= (ln(2), 0) to D= (0, 1) and then from D to B= (-ln(2), 0)" you mean that C is the line from A to D, followed by the line from D to B.

Yes, by Green's theorem, the integral of that function, around the closed path, "A to D to B to A" is 0. But the problem only asks you to integrate for A to D to B, NOT back to A.

The point of the hint is that, since the integral around the closed path is 0, the integral "from A to D to B" must be the negative of the integral from B to A and so equal to the integral from A to B. You will still need to find the path integral from A= (ln 2, 0) to B= (-ln 2, 0) directly. Since y= 0 on that path, it should be relatively easy.