Editing Lists of integrals (section)

==Historical development of integrals==
A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician {{Interlanguage link multi|Meier Hirsch|de}} (also spelled Meyer Hirsch) in 1810.<ref>{{Cite book |last=Hirsch |first=Meyer |url=https://books.google.com/books?id=8IUAAAAAMAAJ |title=Integraltafeln: oder, Sammlung von integralformeln |date=1810 |publisher=Duncker & Humblot |language=de}}</ref> These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician [[David Bierens de Haan]] for his ''[[Tables d'intégrales définies]]'', supplemented by ''[[Supplément aux tables d'intégrales définies]]'' in ca. 1864<!-- no visible date indication by itself. Google books entry states 1864, foreword states 1861, English WP article formerly stated 1862 -->. A new edition was published in 1867 under the title ''[[Nouvelles tables d'intégrales définies]]''.

These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of [[Gradshteyn and Ryzhik]]. In Gradshteyn and Ryzhik, integrals originating from the book by Bierens de Haan are denoted by BI.

Not all [[closed-form expression]]s have closed-form antiderivatives; this study forms the subject of [[differential Galois theory]], which was initially developed by [[Joseph Liouville]] in the 1830s and 1840s, leading to [[Liouville's theorem (differential algebra)|Liouville's theorem]] which classifies which expressions have closed-form antiderivatives. A simple example of a function without a closed-form antiderivative is {{math|''e''<sup>−''x''<sup>2</sup></sup>}}, whose antiderivative is (up to constants) the [[error function]].

Since 1968 there is the [[Risch algorithm]] for determining indefinite integrals that can be expressed in term of [[elementary function]]s, typically using a [[computer algebra system]]. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the [[Meijer G-function]].