Mathematical exploration of integration techniques through substitution, parts, and partial fractions in integral transforms with applications

Abstract

This research paper explores the mathematical foundations and applications of integration techniques substitution, integration by parts, and partial fractions within the framework of integral transforms. These methods play a critical role in solving complex integrals that arise in pure mathematics and applied fields. The study emphasizes the theoretical underpinnings of each technique, demonstrates their applications through rigorous examples, and connects them to integral transforms like the Laplace and Fourier transforms. The exploration highlights the efficiency and flexibility of these techniques in simplifying integrals and transforming mathematical models.