In this article we give an algorithm which produces a basis of the $n$-th de Rham cohomology of the affine smooth hypersurface $f^{-1}\left(t\right)$ compatible with the mixed Hodge structure, where $f$ is a polynomial in $n+1$ variables and satisfies a certain regularity condition at infinity (and hence has isolated singularities). As an application we show that the notion of a Hodge cycle in regular fibers of $f$ is given in terms of the vanishing of integrals of certain polynomial $n$-forms in ${ℂ}^{n+1}$ over topological $n$-cycles on...