An **improper integral** (not to be confused with an indefinite integral) is the limit of a definite integral where the endpoints approach a given value. For a given improper integral to exist, it must converge, which it does if the limit exists. An example would be

This can be solved by taking a limit.

Improper integrals result in infinite values when the series is divergent. For instance,

Improper integrals where both endpoints approach infinity can be solved by breaking them into two improper integrals. For example:

Improper integrals can also be used when over a defined area when there is a vertical asymptote. For example,

## Examples of improper integrals

- Gabriel's Horn, a solid of revolution of an improper integral with a finite value. The shape has infinite surface area but finite volume.
- Gaussian integral
- Laplace transform
- Gamma function