We define the twisted de Rham cohomology and show how to use it to define the
notion of an integral of the form $\int g(x) e^{f(x)}dx$ over an arbitrary ring. We discuss
also a definition of a family of integrals and some properties of the homological
definition of integral. We show how to use the twisted de Rham cohomology in order to
define the Frobenius map on the p-adic cohomology. Finally, we consider two-dimensional
topological quantum field theories with general coefficients.