Method for the Exact Analytical Representation and Evaluation of Special Functions Defined by Non-Elementary Integrals via Cauchy's Mean Value Theorem

This paper introduces a novel computational and analytical framework for representing non-elementary integrals through the application of Cauchy's Mean Value Theorem. By establishing a functional relationship between a target integral

and a selected auxiliary elementary function

, we demonstrate that the value of the integral

can be exactly determined at a specific mean point

. This point

is derived via the Lagrangian geometric projection of the chord. A comprehensive catalog of auxiliary function pairs for fundamental special functions, including Integral Sine, Error Function, and Fresnel Integrals, is provided, utilizing the

 integrand representation.