Editing Integral (section)

=== Other integrals ===
Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including:

* The [[Darboux integral]], which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the [[Riemann integral]]. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being easier to define than Riemann integrals.
* The [[Riemann–Stieltjes integral]], an extension of the Riemann integral which integrates with respect to a function as opposed to a variable.
* The [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes integral]], further developed by [[Johann Radon]], which generalizes both the Riemann–Stieltjes and Lebesgue integrals.
* The [[Daniell integral]], which subsumes the Lebesgue integral and [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes integral]] without depending on [[Measure (mathematics)|measures]].
* The [[Haar integral]], used for integration on locally compact topological groups, introduced by [[Alfréd Haar]] in 1933.
* The [[Henstock–Kurzweil integral]], variously defined by [[Arnaud Denjoy]], [[Oskar Perron]], and (most elegantly, as the gauge integral) [[Jaroslav Kurzweil]], and developed by [[Ralph Henstock]].
* The [[Khinchin integral]], named after [[Aleksandr Khinchin]].
* The [[Itô integral]] and [[Stratonovich integral]], which define integration with respect to [[semimartingale]]s such as [[Wiener process|Brownian motion]].
* The [[Young integral]], which is a kind of Riemann–Stieltjes integral with respect to certain functions of [[Bounded variation|unbounded variation]].
* The [[rough path]] integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both [[semimartingale]]s and processes such as the [[fractional Brownian motion]].
* The [[Choquet integral]], a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953.
* The [[Bochner integral]], a generalization of the Lebesgue integral to functions that take values in a [[Banach space]].