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 $f(x)$ and a selected auxiliary elementary function $g(x)$, we demonstrate that the value of the integral $f(b)$ can be exactly determined at a specific mean point $c \in (0, b)$. This point $c$ is derived via the Lagrangian geometric projection of the chord. A comprehensive catalog of auxiliary function pairs, including normalized Integral Sine, Gaussian Probability, and Fresnel classes, is provided, ensuring exact analytical representation through the $A(t)$ integrand mapping.