  Prompted by an observation about the integral of exponential functions of the
form $f(x)=\lambda\mathrm{e}^{\alpha x}$, we investigate the possibility to
exactly integrate families of functions generated from a given function by
scaling or by affine transformations of the argument using nonlinear
generalizations of quadrature formulae. The main result of this paper is that
such formulae can be explicitly constructed for a wide class of functions, and
have the same accuracy as Newton-Cotes formulae based on the same nodes. We
also show how Newton-Cotes formulae emerge as the linear case of our general
formalism, and demonstrate the usefulness of the nonlinear formulae in the
context of the Pad\'e-Laplace method of exponential analysis.
