  Non-commutative connections of the second type or hom-connections and
associated integral forms are studied as generalisations of right connections
of Manin. First, it is proven that the existence of hom-connections with
respect to the universal differential graded algebra is tantamount to the
injectivity, and that every finitely cogenerated injective module admits a
hom-connection with respect to any differential graded algebra. The bulk of the
paper is devoted to describing a method of constructing hom-connections from
twisted multi-derivations. The notion of a free twisted multi-derivation is
introduced and the induced first order differential calculus is described. It
is shown that any free twisted multi-derivation on an algebra A induces a
unique hom-connection on A that vanishes on the dual basis for the module of
one-forms. To any flat hom-connection \nabla on A one associates a chain
complex, termed a complex of integral forms on A. The canonical cokernel
morphism to the zeroth homology space is called a \nabla-integral. Examples of
free twisted multi-derivations, hom-connections and corresponding integral
forms are provided by covariant calculi on Hopf algebras or quantum groups and
on quantum spaces. The example of a flat hom-connection within the 3D
left-covariant differential calculus on the quantum group SL(2) is described in
full detail. A descent of hom-connections to the base algebra of a faithfully
flat Hopf-Galois extension or a principal comodule algebra is studied. As an
example, a hom-connection on the standard quantum Podles sphere is presented.
In both cases the complex of integral forms is shown to be isomorphic to the de
Rham complex, and the \nabla-integrals coincide with Hopf-theoretic integrals
or invariant (Haar) measures.
