  Given a group $G$ acting cocompactly on a smooth manifold $M$ by deck
transformations, there is an integration map, defined recently by Kato,
Kishimoto and Tsutaya, from the top-degree bounded de Rham cohomology of $M$ to
the coinvariants $\ell^\infty(G)_G$. We generalize its definition and show that
it is an isomorphism. In the presence of boundary, a relative version of
bounded de Rham cohomology is considered.
