I am supposed to do the integral ∫(1/Ω)dl, where dl is the vector pointing along the loop and Ω is the distance from the origin to a point on the loop. Let region 1 be from x=a to x=b region 2: outer semicircle of radius b region 3: x=-b to x=-a region 4: inner semicircle of radius a. Let's just look at regions 1 and 3: in these regions, dl=dx x, x is the unit vector. dx x is a positive quantity, since it is parallel (not antiparallel) to the arrows in the above figure drawn in these regions (my path of integration). Also in regions 1 and 3, Ω=x. Thus i set up the integral in region 1 as: x∫(1/x)dx , with limits from a->b and in region 3: x∫(1/x)dx , with limits from -b->-a. However, when I add these two results, they cancel, but the solution says they should add. I am having same problem with regions 2 and 4 (they should cancel but in my case they add). Thus I must probably be confused with signs/directions, but i do not see where my error is. Help is much appreciated!