  In this paper we will study integrability of distributions whose primitives
are left regulated functions and locally or globally integrable in the
Henstock--Kurzweil, Lebesgue or Riemann sense. Corresponding spaces of
distributions and their primitives are defined and their properties are
studied. Basic properties of primitive integrals are derived and applications
to systems of first order nonlinear distributional differential equations and
to an $m$th order distributional differential equation are presented. The
domain of solutions can be unbounded, as shown by concrete examples.
