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.

翻译：受形如 $f(x)=\lambda\mathrm{e}^{\alpha x}$ 的指数函数积分的观察启发，本文研究了通过求积公式的非线性推广，精确积分由给定函数经缩放或自变量仿射变换生成的函数族的可能性。本文主要结果表明，此类公式可针对广泛的函数类显式构造，并具有与基于相同节点的牛顿-柯特斯公式同等的精度。同时，我们揭示了牛顿-柯特斯公式如何作为本一般化形式体系的线性特例出现，并论证了非线性公式在指数分析中的帕德-拉普拉斯方法中的应用价值。