Improper Integral

An improper integral is a definite integral defined by a limit of proper integrals when the interval is unbounded or the integrand is unbounded near an endpoint or an interior point. The integrand must be Riemann integrable on the truncated intervals. The improper integral converges when all required limits exist and are finite. At an interior singularity, the two one-sided limits must exist separately (Hunter 2014, pp. 257-259).

The change of variables theorem can remove an integrable endpoint singularity or replace an infinite interval by a finite one. It preserves the corresponding improper limit, so it does not turn a divergent improper integral into a convergent one.

For example, the integral

is an improper integral. Some such integrals can sometimes be computed by replacing infinite limits with finite values

and then taking the limit as ,

Improper integrals of the form

with one infinite limit can be expressed on finite intervals by transforming both the integrand and the limits. For the reciprocal substitution, assume that when , or that when , so the interval does not cross zero. Let

and

These identities are interpreted using the corresponding one-sided limits at zero. If for sufficiently large and some , the transformed integrand is bounded near zero.

For finite and lower-endpoint behavior proportional to with , let

and

For upper-endpoint behavior proportional to with , let

and

For an integrand that decays exponentially as , let

and