  We define a non-absolutely convergent integration on integral currents of
dimension 1 in Euclidean space. This integral is closely related to the
Henstock-Kurzweil and Pfeffer Integrals. Using it, we prove a generalized
Fundamental Theorem of Calculus on these currents. A detailed presentation of
Henstock-Kurzweil Integration is given in order to make the paper accessible to
non-specialists.
