View source for Integral

{{also|INTEGRAL|Integral|intégral|integrál}}
==English==

===Etymology===
{{bor+|en|frm|integral}}, from {{der|en|ML.|integrālis}}, from {{der|en|la|integer|t=entire}}; see [[integer]].

===Pronunciation===
* '''Noun'''
* {{enPR|ĭnʹtĭ-grəl|a=UK}}, {{IPA|en|/ˈɪntɪɡɹ(ə)l/}}
* {{enPR|ĭnʹtə-grəl|a=US}}, {{IPA|en|/ˈɪntəɡɹəl/|/ɪnˈtɛɡɹəl/}}
* {{audio|en|en-uk-integral-noun.ogg|a=UK}}
* '''Adjective'''
* {{IPA|en|/ˈɪntɪɡr(ə)l/|/ɪnˈtɛɡr(ə)l/|a=UK}}
* {{enPR|ĭn-tĕgʹrəl|a=US}}, {{IPA|en|/ˈɪntəɡɹəl/|/ɪnˈtɛɡɹəl/}}
* {{audio|en|en-uk-integral-adjective.ogg|a=UK}}

===Adjective===
{{en-adj}}

# [[constitute|Constituting]] a [[whole]] together with other [[part]]s or [[factor]]s; not [[omittable]] or [[removable]]
#: {{syn|en|immanent|inherent|necessary|Thesaurus:intrinsic}}
#* {{RQ:South Twelve Sermons|passage=Ceasing to do evil, and doing good, are the two great '''integral''' parts that complete this duty.}}
# {{lb|en|mathematics}} Of, pertaining to, or being an [[integer]].
# {{lb|en|mathematics}} Relating to {{l|en|integration||the process of finding the [[integral#Noun|integral [noun]]] of a [[function]]}}.
# {{lb|en|algebra|[[commutative algebra]]|of a [[ring]] [[element]] in a [[ring]] <math>B</math> relative to a [[subring]] <math>A</math>}} Being the [[root]] of some [[monic]] [[polynomial]] in <math>A</math>.
#: {{coordinate terms|en|integral element|algebraic}}
# {{lb|en|obsolete}} [[whole|Whole]]; [[undamaged]].
#* {{RQ:Bacon Sylva Sylvarum|passage=A local motion keepeth bodies '''integral'''.}}

====Derived terms====
{{col|en|integral brick|integral closure|integral cosmology|integral cuboid|integral current|integral curvature|integral curve|integral domain|integral drawing|integral ecology|integral element|integral energy|integral extension|integral fast reactor|integral field unit|integral function|integral geometry|integral graph|integral homology group|integral kernel|integral membrane protein|integral politics|integral polygedron|integral polynomial|integral post-metaphysics|integral psychology|integral transformative practice|integral yoga|integrity|semi-integral|integral number}}

====Related terms====
{{col|en|integer|integrity|integrous<qq:very rare>}}

====Translations====
{{trans-top|constituting a whole together with other parts or factors; not omittable or removable}}
* Bashkir: {{t|ba|айырылғыһыҙ}}
* Bulgarian: {{t+|bg|неразделен}}, {{t+|bg|цялостен}}
* Chinese:
*: Cantonese: {{t|yue|完整嘅|tr=jyun4 zing2 ge3}}
*: Mandarin: {{t|cmn|完整的|tr=wánzhěng de}}
* Czech: {{t|cs|nedílný}}
* Dutch: {{t+|nl|integraal}}
* Esperanto: {{t+|eo|integra}}
* Finnish: {{t+|fi|olennainen}}, {{t|fi|kokonaisuuteen kuuluva}}
* Galician: {{t|gl|integral|m|f}}
* German: {{t+|de|integral}}, {{t|de|[[ein]] [[Ganzes]] [[ausmachend]]}}, {{t|de|[[zur]] [[Erhaltung]] [[des]] [[Ganzen]] [[unentbehrlich]]}} {{qualifier|descriptive}}
* Greek: {{t+|el|αναπόσπαστος}}
* Hebrew: {{t+|he|אִינְטֶגְרָלִי|tr=integráli}}
* Kazakh: {{t|kk|интегралды}}, {{t|kk|интегралдық}}
* Khmer: {{t|km|ដែលទាំងស្រុង|tr=dael teang srong}}
* Manx: {{t|gv|slane}}
* Persian: {{t-needed|fa}}
* Polish: {{t+|pl|jednolity}}, {{t+|pl|integralny}}
* Portuguese: {{t+|pt|integral}}
* Romanian: {{t+|ro|integral}}
* Russian: {{t+|ru|неотъе́млемый}}, {{t+|ru|недели́мый}}, {{t+|ru|неотдели́мый}}
* Swedish: {{t+|sv|hel}}, {{t+|sv|enhetlig}}
{{trans-bottom}}

{{trans-top|math: of, pertaining to, or being an integer}}
* Arabic: {{t+|ar|تَكَامُل|m}}
* Bashkir: {{t|ba|бөтөн һанлы}}
* Czech: {{t|cs|celočíselný}}
* Dutch: {{t+|nl|geheel}}
* Esperanto: {{t|eo|entjera}}
* Finnish: {{t+|fi|kokonaisluku|alt=kokonaisluku-}}
* French: {{t+|fr|intégral}}
* German: {{t+|de|ganzzahlig}}
* Greek: {{t+|el|ακέραιος|m}}
* Kazakh: {{t|kk|интегралды}}, {{t|kk|интегралдық}}
* Manx: {{t|gv|slane-earrooagh}}
* Portuguese: {{t+|pt|inteiro}}
* Romanian: {{t+|ro|integral}}
* Russian: {{t+|ru|целочи́сленный}}
* Spanish: {{t+|es|entero}}
* Swedish: compounds with {{t+|sv|heltal}}
* Tagalog: {{t|tl|buumbilangin}}
{{trans-bottom}}

{{trans-top|math: relating to integration}}
* Bulgarian: {{t+|bg|интегрален}}
* Esperanto: {{t|eo|integrala}}
* Finnish: {{t+|fi|integraali|alt=integraali-}}
* French: {{t+|fr|intégral}}
* Galician: {{t|gl|integral}}
* Kazakh: {{t|kk|интегралды}}, {{t|kk|интегралдық}}
* Spanish: {{t+|es|integral}}
{{trans-bottom}}

{{checktrans-top}}
* Hungarian: {{t+check|hu|integrál}} (2)
* Manx: {{t-check|gv|co-hymagh}}
{{trans-bottom}}

===Noun===
{{wikipedia}}
{{en-noun}}

# {{lb|en|mathematics}} One of the two fundamental [[operations]] of [[calculus]] (the other being [[differentiation]]), whereby a [[function]]'s [[displacement]], [[area]], [[volume]], or other qualities arising from the study of [[infinitesimal]] change are quantified, usually defined as a [[limit|limiting]] process on a [[sequence]] of [[partial sum]]s. Denoted using a [[long s]]: [[∫]], or a variant thereof.
#: {{ux|en|The '''integral''' of a [[univariate ]][[real-valued]] function is the area under its curve; but be warned! Not all functions are [[integrable]]!}}
## {{lb|en|specifically}} Any of several [[analysis|analytic]] [[formalization]]s of this operation: the [[Riemann integral]], the [[Lebesgue integral]], etc.
# {{lb|en|mathematics}} A [[definite integral]]: the result of the [[application]] of such an operation onto a function and a suitable [[subset]] of the function's [[domain]]: either a [[number]] or [[positive]] or [[negative]] [[infinity]]. In the former case, the integral is said to be [[finite]] or to [[converge]]; in the latter, the integral is said to [[diverge]]. In notation, the domain of integration is indicated either below the sign, or, if it is an [[interval]], with its endpoints as sub- and super-scripts, and the function being integrated forming part of the [[integrand]] (or, generally, [[differential form]]) appearing in front of the integral sign.
#: {{ux|en|The '''integral''' of <math>\frac{1}{x}</math> on <math>[\frac{1}{2}, 1]</math> is <math>\ln(2)</math>, but the '''integral''' of the same function on <math>(0, 1]</math> diverges. In notation, <math>\int_\frac{1}{2}^1\frac{1}{x} dx = \ln(2)</math>, but <math>\int_0^1\frac{1}{x} dx = \infty</math>.}}
#: {{ux|en|Stokes' theorem relates the '''integral''' over a surface of the [[curl]] of a [[vector field]] to a [[line integral|line '''integral''']] around the boundary of that surface: <math>    \int_S \nabla\times \mathbf{F}\cdot d\mathbf{S} = \int_{\partial S} \mathbf{F}\cdot d\mathbf{r},
</math>.}}
# {{lb|en|mathematics}} An [[indefinite integral]]: the result of the application of such an operation onto a function together with an [[indefinite]] domain, yielding a function; a function's [[antiderivative]];
#: {{syn|en|antiderivative|indefinite integral|mul:∫}}
#: {{ant|en|derivative}}
#: {{ux|en|The '''integral''' of <math>x^2</math> is <math>\frac{x^3}{3}</math> plus a constant.}}
## {{lb|en|mathematics|historical or obsolete}} The [[fluent]] of a given [[fluxion]] in [[Newtonian]] calculus.

====Derived terms====
{{col|en|definite integral|half-integral|improper integral|indefinite integral|integral calculus|integral equation|integral theory|integral transform|integration|Riemann integral|Choquet integral|iterated integral|Jackson integral|Sugeno integral|Riemann-Stieltjes integral|Lebesgue integral|Barnes integral|contour integral|Hellinger integral|line integral|logarithmic integral|Mellin-Barnes integral|ordered integral domain|particular integral|path integral formalism|surface integral|Volterra integral equation|volume integral}}

====Related terms====
{{col|en|integer}}

====Translations====
{{trans-top|limit of sums}}
* Armenian: {{t+|hy|ինտեգրալ}}
* Belarusian: {{t|be|інтэгра́л|m}}
* Bulgarian: {{t|bg|интегра́л|m}}
* Catalan: {{t+|ca|integració|f}}
* Chinese:
*: Cantonese: {{t|yue|積分|tr=zik1 fan1}}
*: Hakka: {{t|hak|chit-fûn-ho̍k}}
*: Mandarin: {{t+|cmn|積分|tr=jī fēn}}
* Czech: {{t+|cs|integrál|m}}
* Dutch: {{t+|nl|integraal}}
* Esperanto: {{t+|eo|integralo}}, {{t|eo|integro}}
* Finnish: {{t+|fi|integraali}}
* French: {{t+|fr|intégrale|f}}
* Galician: {{t|gl|integral|f}}
* Georgian: {{t|ka|ინტეგრალი}}
* German: {{t+|de|Integral|n}}
* Greek: {{t+|el|ολοκλήρωμα|n}}
* Hebrew: {{t+|he|אִינְטֶגְרָל|m|tr=integrál}}, {{t+|he|אַסְכֶּמֶת|f|tr=askémet}} {{qualifier|Academy of the Hebrew Language}}
* Hungarian: {{t+|hu|integrál}}
* Icelandic: {{t|is|heildun|f}}, {{t|is|tegrun|f}}
* Italian: {{t+|it|integrale|m}}
* Japanese: {{t+|ja|積分|tr=せきぶん, sekibun}}
* Kazakh: {{t|kk|интеграл}}
* Khmer: {{t+|km|អាំងតេក្រាល}}
* Korean: {{t+|ko|적분}}
* Kyrgyz: {{t|ky|интеграл}}
* Malay: {{t|ms|kamiran}}
* Manx: {{t|gv|co-hym|m}}
* Mongolian: {{t+|mn|интеграл}}
* Norwegian: {{t|no|integral|n}}
* Polish: {{t+|pl|całka|f}}, {{t+|pl|całka oznaczona|f}}
* Portuguese: {{t+|pt|integral|f}}
* Romanian: {{t+|ro|integrală|f}}
* Russian: {{t+|ru|интегра́л|m|tr=intɛgrál}}
* Serbo-Croatian:
*: Cyrillic: {{t|sh|интѐгра̄л|m}}
*: Roman: {{t+|sh|intègrāl|m}}
* Spanish: {{t+|es|integral|f}}
* Sundanese: {{t|su|integral}}
* Swedish: {{t+|sv|integral|c}}
* Tagalog: {{t|tl|lagom}}
* Thai: {{t|th|ปริพันธ์}}
* Turkish: {{t+|tr|integral}}
* Ukrainian: {{t+|uk|інтегра́л|m}}
* Vietnamese: {{t+|vi|tích phân}}
{{trans-bottom}}

{{trans-see|antiderivative}}

===Anagrams===
* {{anagrams|en|a=aegilnrt|Triangle|alerting|altering|relating|tanglier|teraglin|triangle}}

{{C|en|Functions}}
{{cln|en|heteronyms}}
{{C|en|Calculus}}

==Albanian==

===Noun===
{{sq-noun|m}}

# {{lb|sq|mathematics}} [[#English|integral]] (''relating to [[integration]]'')

===Further reading===
*{{R:sq:Fjalorthi}}
*{{R:FGSS}}

==Catalan==

===Etymology===
{{bor+|ca|ML.|integrālis}}, from {{der|ca|la|integer}}.

===Pronunciation===
* {{ca-IPA}}

===Adjective===
{{ca-adj}}

# {{l|en|integral}}
# {{gloss|of bread, etc.}} [[wholegrain]], [[wholemeal]]

====Related terms====
{{col3|ca|enter|integració|integrar|íntegre}}

===Noun===
{{ca-noun|f}}

# {{lb|ca|mathematics}} {{l|en|integral}}

====Derived terms====
* {{l|ca|integral indefinida}}

===Further reading===
* {{R:ca:IEC2}}

==Indonesian==

===Etymology===
{{bor+|id|nl|integraal|t=integral}}, from {{der|id|fr|intégral}}, from {{der|id|ML.|integrālis}}, from {{der|id|la|integer||entire}}.

===Pronunciation===
{{id-pr|integrāl}}

===Adjective===
{{id-adj|head=intêgral}}

# {{l|en|integral}}:
## constituting a whole together with other parts or factors; not omittable or removable
##: {{syn|id|bulat|utuh|sempurna}}
## {{lb|id|mathematics}} relating to integration (“the process of finding the integral [noun] of a function”)

===Noun===
{{id-noun|head=intêgral}}

# {{lb|id|mathematics}} {{l|en|integral}}

====Related terms====
{{col4|id|
| integrasi
|}}

===Further reading===
* {{R:KBBI Daring}}

==Middle French==

===Etymology===
Borrowed from {{bor|frm|ML.|integrālis}}, from {{der|frm|la|integer}}.

===Adjective===
{{frm-adj|integrale|integraux|integrales}}

# {{l|en|integral}}, necessary to the function of the whole
# [[whole]]; [[entire]]

====Descendants====
* {{desc|fr|intégral}}

===References===
* {{R:DMF|intégral|alt=integral}}

==Portuguese==

===Etymology===
Borrowed from {{bor|pt|ML.|integrālis}}, from {{der|pt|la|integer|t=entire; untouched}}.

===Pronunciation===
{{pt-IPA}}
* {{rhymes|pt|al|aw|s=3}}
* {{hyphenation|pt|in|te|gral}}

===Adjective===
{{pt-adj}}

# {{l|en|integral}}; [[whole]]; [[entire]]
# {{lb|pt|of food}} [[whole]] {{gloss|from which none of its constituents has been removed}}

====Quotations====
{{seeCites|pt}}

====Synonyms====
* {{sense|whole}} {{l|pt|completo}}, {{l|pt|íntegro}}, {{l|pt|inteiro}}, {{l|pt|intacto}}, {{l|pt|total}}

====Derived terms====
* {{l|pt|integralmente}}

===Noun===
{{pt-noun|f|g2=m}}

# {{lb|pt|mathematics}} {{l|en|integral}} {{gloss|limits of sums}}
# {{lb|pt|mathematics}} [[antiderivative]]
#: {{syn|pt|antiderivada}}

====Quotations====
{{seeCites|pt}}

====Related terms====
{{col3|pt
|integrado
|integrar
|íntegro
}}

===Further reading===
* {{R:pt:Priberam}}

==Romanian==

===Etymology===
{{bor+|ro|fr|intégral}}, {{bor|ro|ML.|integrālis}}.

===Pronunciation===
* {{IPA|ro|/in.teˈɡral/}}

===Adjective===
{{ro-adj}}

# {{l|en|integral}}
#: {{syn|ro|întreg|complet}}

====Declension====
{{ro-adj-4}}

====Related terms====
* {{l|ro|integru}}

==Spanish==

===Etymology===
{{bor+|es|ML.|integrālis}}, from {{der|es|la|integer|t=entire}}.

===Pronunciation===
{{es-pr|+<audio:LL-Q1321 (spa)-AdrianAbdulBaha-integral.wav<a:Colombia>>}}

===Adjective===
{{es-adj}}

# {{l|en|integral}}
# [[whole]]
# [[brown]] {{q|rice}}
# [[wholegrain]]

====Derived terms====
{{col|es|arroz integral|cálculo integral|integralidad|integralmente|pan integral|parte integral}}

====Related terms====
{{col|es|íntegro}}

===Noun===
{{es-noun|f}}

# {{lb|es|mathematics}} {{l|en|integral}}

===Further reading===
* {{R:es:DRAE}}

==Swedish==
{{wikipedia|lang=sv}}

===Etymology===
{{rfe|sv}}

===Pronunciation===
* {{IPA|sv|/ɪntɛˈɡrɑːl/}}
* {{hyph|sv|in|te|gral}}
* {{rhymes|sv|ɑːl|s=3}}

===Noun===
{{sv-noun|c}}

# {{lb|sv|mathematics}} an {{l|en|integral}}

====Declension====
{{sv-infl-noun-c-er}}

====Related terms====
* {{l|sv|integrand}}
* {{l|sv|integration}}
* {{l|sv|integrera}}

===References===
* {{R:svenska.se|so}}
* {{R:svenska.se|saol}}
* {{R:svenska.se|saob}}

===Anagrams===
* {{anagrams|sv|a=aegilnrt|triangel}}

==Tagalog==

===Etymology===
{{bor+|tl|es|integral}}.

===Pronunciation===
{{tl-pr|integrál}}

===Adjective===
{{tl-adj|integrál|b=+}}

# {{l|en|integral}}
#: {{syn|tl|buo|kompleto}}

====Related terms====
{{col|tl|integrasyon|integrante}}

==Turkish==

===Etymology===
{{bor+|tr|fr|intégral}}.

===Pronunciation===
* {{IPA|tr|[in.tɛɟ.ˈɾalʲ]}}

===Noun===
{{tr-noun|i|ler}}

# {{lb|tr|mathematics}} {{l|en|integral}}
#: <math>\int_a^b \! f(x)\,dx \,</math>

====Declension====
{{tr-infl-noun-c|e|poss=1}}