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

Distributions, their primitives and integrals with applications to differential equations