Editing Constant of integration (section)

{{Short description|Constant expressing ambiguity from indefinite integrals}}
{{More citations needed|date=August 2020}}
In [[calculus]], the '''constant of integration''', often denoted by <math>C</math> (or <math>c</math>), is a [[constant term]] added to an [[antiderivative]] of a function <math>f(x)</math> to indicate that the [[indefinite integral]] of <math>f(x)</math> (i.e., the [[Set (mathematics)|set]] of all antiderivatives of <math>f(x)</math>), on a [[connected set|connected domain]], is only defined [[up to]] an additive constant.<ref>{{cite book | last=Stewart | first=James | author-link=James Stewart (mathematician) | title=Calculus: Early Transcendentals | publisher=[[Brooks/Cole]] | edition=6th | year=2008 | isbn=978-0-495-01166-8 | url-access=registration | url=https://archive.org/details/calculusearlytra00stew_1 }}</ref><ref>{{cite book | last1=Larson | first1=Ron | author-link=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=[[Brooks/Cole]] | edition=9th | year=2009 | isbn=978-0-547-16702-2}}</ref><ref>{{Cite web|title=Definition of constant of integration {{!}} Dictionary.com|url=https://www.dictionary.com/browse/constant-of-integration|access-date=2020-08-14|website=www.dictionary.com|language=en}}</ref> This constant expresses an ambiguity inherent in the construction of antiderivatives.

More specifically, if a function <math>f(x)</math> is defined on an [[interval (mathematics)|interval]], and <math>F(x)</math> is an antiderivative of <math>f(x),</math> then the set of ''all'' antiderivatives of <math>f(x)</math> is given by the functions <math>F(x) + C,</math> where <math>C</math> is an arbitrary constant (meaning that ''any'' value of <math>C</math> would make <math>F(x) + C</math> a valid antiderivative). For that reason, the indefinite integral is often written as <math display="inline">\int f(x) \, dx = F(x) + C,</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Constant of Integration|url=https://mathworld.wolfram.com/ConstantofIntegration.html|access-date=2020-08-14|website=mathworld.wolfram.com|language=en}}</ref> although the constant of integration might be sometimes omitted in [[lists of integrals]] for simplicity.