  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.
