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 the exact evaluation of non-elementary integrals. By synthesizing Cauchy's Mean Value Theorem with a Lagrangian geometric projection, we establish the fundamental identity f(b) = b · A(c). A rigorous procedure for determining the Lagrangian Geometric Invariant point c through the characteristic equation of the auxiliary mapping is established. We demonstrate that integration can be reduced to a direct algebraic product, providing exact analytical representations for primary special functions, including transcendental cases resolved via the Lambert W function.