  We study integral operators related to a regularized version of the classical
Poincar\'e path integral and the adjoint class generalizing Bogovski\u{\i}'s
integral operator, acting on differential forms in $R^n$. We prove that these
operators are pseudodifferential operators of order -1. The Poincar\'e-type
operators map polynomials to polynomials and can have applications in finite
element analysis. For a domain starlike with respect to a ball, the special
support properties of the operators imply regularity for the de Rham complex
without boundary conditions (using Poincar\'e-type operators) and with full
Dirichlet boundary conditions (using Bogovski\u{\i}-type operators). For
bounded Lipschitz domains, the same regularity results hold, and in addition we
show that the cohomology spaces can always be represented by $C^\infty$
functions.
