The Second Mean Value Theorem for Integrals

This article discusses the Second Mean Value Theorem for integrals by presenting a comprehensive mathematical proof using a deductive-mathematical approach that involves the Extreme Value Theorem and the Comparison Theorem. Given a continuous function and an integrable function that does not change sign on the interval , it is proven that there exists at least one point such that: \[ \int_a^b f(x)g(x)\,dx = f(c) \int_a^b g(x)\,dx \] The article also provides various examples of the theorem’s application, including numerical computations using the Newton-Raphson method to determine the value of in certain cases. In addition, case studies are presented that link the theorem to modeling in probability, economics, and engineering, thereby demonstrating its relevance in data analysis and dynamic systems. The results of this study not only enrich the theoretical foundation of integral analysis but also offer practical contributions to problem solving in various disciplines.