  Scalar field cosmological inflation has a first integral relating the Hubble
function and the lagrangian of the scalar field(s), which is known under the
names of "Hamilton-Jacobi approach" or "superpotential equation". Here we
exploit the simplicity of this superpotential equation and use it as an
alternative but equivalent cosmological evolution equation during inflation,
replacing the Friedman-Robertson-Walker (FRW) equations. It turns out that all
inflationary observables can be calculated directly from its solution (the
superpotential). Further, the superpotential equation allows for a simple and
direct calculation of the slow-roll expansion to arbitrary order and, in many
cases, for an exact determination of the slow-roll attractor. It also allows
for a power series expansion in the inflaton field which permits to estimate
the radius of convergence of the slow-roll expansion. We consider several
examples of single-field inflationary models to demonstrate the simplicity and
usefulness of the method.
