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This fourth volume, Advanced Numerical and Algorithmic Approaches for Irreducible Integrals, is intended for students, researchers, and engineers dealing with integrals that cannot be solved analytically. Through a rigorous framework, it presents error bounds, estimates of maximum achievable precision, and methods for determining the optimal number of intervals required. Each numerical technique is illustrated with an algorithm, a Python implementation, and exercises grounded in diverse application domains such as biology, finance, and telecommunications. By combining mathematical theory, computational tools, and real-world case studies, the book offers a clear methodological framework for assessing and mastering the precision of numerical methods. This volume completes a four-part series dedicated to both analytical and numerical integration, delivering a structured, comprehensive, and practically oriented vision of integral calculus in the sciences and engineering.