A Unified Substitution Method for Integration

This paper introduces the Unified Substitution Method (USM), a systematic framework for integrating expressions involving inverse trigonometric functions, radicals, and related forms. By synthesizing classical techniques -- such as Euler substitutions, hyperbolic parameterizations, and complex exponential identities -- into a single cohesive strategy, the USM streamlines the evaluation of integrals that traditionally require fragmented, case-specific approaches. Central to the method are novel Euler-like identities that rigorously unify trigonometric, hyperbolic, and algebraic substitutions while preserving domain consistency. These transformations reduce complex integrands to rational or polynomial-like forms, bypassing ad hoc substitutions and memorization of standard formulas. Demonstrations include integrals resistant to standard Computer Algebra Systems (CAS), such as those mixing inverse trigonometric functions, and comparisons with classical Euler substitutions reveal that the latter emerge as special cases within the USM framework. Crucially, the USM extends beyond Euler substitutions by systematically handling diverse forms -- including quadratic radicals with linear shifts, inverse trigonometric integrands, and mixed-function compositions -- under a single principle. The method enhances computational efficiency, clarifies domain-specific sign choices, and serves as a robust educational and practical tool, offering a unified alternative to historically disjointed techniques in integral calculus.