Generalized improper integral definition for infinite limit

Abstract

A generalization of the definition of a one-dimensional improper integral with an infinite limit is presented. The new definition extends the range of valid integrals to include integrals which were previously considered to not be integrable. This definition is shown to preserve linearity and uniqueness. Integrals which are valid under the conventional definition have the same value under the new definition. Criteria for interchanging the order of integration and differentiation, and for interchanging the order of integration with a second integration, are determined. Examples are provided. A restriction on changing the variable of integration using integration by substitution with the new definition is demonstrated.