An improper integral is an integral of one of two types: either the interval over which the interval is taken is unbounded, or the integrand becomes unbounded in a neighborhood of one or more points in the interval (including the endpoints). The common thread in these two types of integrals is that both have the potential to be infinite.

The first type of integral is an integral over either a half-infinite interval, i.e. from a finite value a to infinity or from negative infinity to a, or over the interval from negative infinity to positive infinity. Let us first consider the first of these possibilities. Suppose f is a function that is defined for all x greater than or equal to a. We are trying to measure an area whose "horizontal dimension" is infinite. The mathematical method for doing computations involving infinity is to take...