(untitled) (Patan Prospective Journal, 2025)
This research paper is about the convergence and divergence of improper integrals, or integrals with infinite limits of integration, or integrands with singularities. The purpose of the research is to mathematically rigorously study under what conditions improper integrals converge or diverge and demonstrate the usefulness of these tendencies, particularly in probability theory, physics, and engineering. Detailed analysis and solved examples illustrate the use of convergence and divergence for ascertaining area, computing probabilities, and solving differential equations. The paper also compares different methods of determining convergence and divergence, including the comparison tests, the limit comparison tests, and the Cauchy principal value.