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 f(x)f of x 𝑓(𝑥) and a selected auxiliary elementary function g(x)g of x 𝑔(𝑥) , we demonstrate that the value of the integral f(b)f of b 𝑓(𝑏) can be exactly determined at a specific mean point c∈(0,b)c is an element of open paren 0 comma b close paren 𝑐∈(0,𝑏) . This point cc 𝑐 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 A(t)cap A open paren t close paren 𝐴(𝑡) integrand representation.