  For a manifold M we define a structure on the group action of Diff(M) on the
smooth functions on M which reduces to the usual differential geometry upon
differentiation at zero along the one-parameter groups of Diff(M). This
``integrated differential geometry'' generalises to all group actions on
associative algebras, including noncommutative ones, and defines an
``integrated de Rham cohomology,'' which provides a new set of invariants for
group actions. We calculate the first few integrated de Rham cohomologies for
two examples;- a discrete group action on a commutative algebra, and a
continuous Lie group action on a noncommutative matrix algebra.
