In mathematical applications to some fundamental problems in Physics and Mechanics one needs to perform integration over the "solution space" of a given non-linear PDE (Feynman path integral, etc.). It seems that this goal cannot be reached by standard measure theory methods. The first part of the course aims to show that the integral is actually a cohomological concept and, in the simplest case (integral over a smooth manifold), an aspect of the theory of de Rham cohomology. The main techniques of computation of de Rham cohomology will be introduced on the base of differential calculus thus avoiding the standard use of algebraic topology. Among these techniques a central role is played by the differential version of the Leray-Serre spectral sequence. Such a sequence is not only important in its own but it is also the most simple example of a C-spectral sequence, which is a key notion in Secondary Calculus.