Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Under suitable circumstances, forming the integral of a differential form may be understood as passing to the cohomology-equivalence class of that differential form in a suitable chain complex.
Examples of this appear in the constructions discussed at Lie integration and at BV-BRST formalism, where cohomological integration is used as a way to formalize the idea of the path integral. (See at The BV-complex and homological (path-)integration)
While the idea has been around (as witnessed by the references at Lie integration and BV-BRST formalism) a comprehensive and dedicated theory, or a published account thereof, currently seems to be missing. But see the References below.