text: When you walk from a point  to a point  along a line  (without changing directions) your net displacement and total distance covered are both equal to the length of said line . If you then return to point  (without changing directions) then your net displacement is zero while your total distance covered is . This example captures the essential idea behind the inexact differential in one dimension. Note that if we allowed ourselves to change directions, then we could take a step forward and then backward at any point in time in going from  to  and in-so-doing increase the overall distance covered to an arbitrarily large number while keeping the net displacement constant.
  Reworking the above with differentials and taking  to be along the -axis, the net distance differential is , an exact differential with antiderivative . On the other hand, the total distance differential is , which does not have an antiderivative. The path taken is  where there exists a time  such that  is strictly increasing before  and strictly decreasing afterward. Then  is positive before  and negative afterward, yielding the integrals,
  exactly the results we expected from the verbal argument before.