integral

See also: INTEGRAL, Integral, intégral, and integrál

English

Etymology

Borrowed from Middle French integral, from Medieval Latin integrālis, from Latin integer (“entire”); see integer.

Pronunciation 1

• (Received Pronunciation) IPA^(key): /ˈɪntɪɡr(ə)l/, /ɪnˈtɛɡr(ə)l/
• (General American) enPR: ʹĭn-tə-grəl, enPR: ĭn-tĕgʹrəl, IPA^(key): /ˈɪntəɡɹəl/, /ɪnˈtɛɡɹəl/

• Audio (UK):

  (file)

• Audio (US):

  (file)

Adjective

integral (comparative more integral, superlative most integral)

1. Constituting a whole together with other parts or factors; not omittable or removable.

   Synonyms: immanent, inherent, necessary; see also Thesaurus:intrinsic

   • 1692–1717, Robert South, Twelve Sermons Preached upon Several Occasions, volume (please specify |volume=I to VI), London:

     Ceasing to do evil, and doing good, are the two great integral parts that complete this duty.

   • 2021 December 17, Vanessa Friedman, “The Verdict on the Elizabeth Holmes Trial Makeover”, in The New York Times‎^[1], New York, N.Y.: The New York Times Company, →ISSN, →OCLC, archived from the original on 12 April 2022:

     When the verdict comes down, the transformation of the wunderkind founder of Theranos from black-clad genius to besuited milquetoast will be an integral part of the story.

2. (mathematics) Of, pertaining to, or being an integer.
3. (mathematics) Relating to integration (“the process of finding the integral [noun] of a function”).
4. (algebra, commutative algebra, of a ring element in a ring $B$ relative to a subring $A$) Being the root of some monic polynomial in $A$.

   Coordinate terms: integral element, algebraic

5. (obsolete) Whole; undamaged.
   • 1627 (indicated as 1626), Francis [Bacon], “(please specify the page, or |century=I to X)”, in Sylua Syluarum: Or A Naturall Historie. In Ten Centuries. […], London: […] William Rawley […]; [p]rinted by J[ohn] H[aviland] for William Lee […], →OCLC:

     A local motion keepeth bodies integral.

Derived terms

• differintegral
• 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
• integralism
• integralist
• integrality
• integral kernel
• integrally
• integral membrane protein
• integralness
• integral number
• integral politics
• integral polygedron
• integral polynomial
• integral post-metaphysics
• integral psychology
• integral transformative practice
• integral yoga
• integraph
• integrity
• nonintegral
• semi-integral
• unintegral

Related terms

• integer
• integrity
• integrous (very rare)

Translations

constituting a whole together with other parts or factors; not omittable or removable

• Bashkir: айырылғыһыҙ (ayırılğıhıð)
• Bulgarian: неразделен (bg) (nerazdelen), цялостен (bg) (cjalosten)
• Chinese:

  Cantonese: 完整嘅 (jyun⁴ zing² ge³)

  Mandarin: 完整的 (wánzhěng de)

• Czech: nedílný
• Dutch: integraal (nl)
• Esperanto: integra (eo)
• Finnish: olennainen (fi), kokonaisuuteen kuuluva
• French: intégral (fr)
• Galician: integral (gl) m or f
• German: integral (de), ein Ganzes ausmachend, zur Erhaltung des Ganzen unentbehrlich (descriptive)
• Greek: αναπόσπαστος (el) (anapóspastos)
• Hebrew: אִינְטֶגְרָלִי (he) (integráli)
• Hindi: समाकलन m (samāklan)
• Kazakh: интегралды (ıyntegraldı), интегралдық (ıyntegraldıq)
• Khmer: ដែលទាំងស្រុង (dael teang srong)
• Manx: slane
• Persian: please add this translation if you can
• Polish: jednolity (pl), integralny (pl)
• Portuguese: integral (pt)
• Romanian: integral (ro)
• Russian: неотъе́млемый (ru) (neotʺjémlemyj), недели́мый (ru) (nedelímyj), неотдели́мый (ru) (neotdelímyj)
• Spanish: integral (es), íntegro (es)
• Swedish: hel (sv), enhetlig (sv)
• Turkish: ayrılmaz

math: of, pertaining to, or being an integer

• Arabic: صَحِيح (ar) m (ṣaḥīḥ)
• Bashkir: бөтөн һанлы (bütün hanlı)
• Bulgarian: целочислен (celočislen)
• Czech: celočíselný
• Dutch: geheel (nl)
• Esperanto: entjera
• Finnish: kokonaisluku- (fi)
• French: entier (fr)
• German: ganzzahlig (de)
• Greek: ακέραιος (el) m (akéraios)
• Hindi: समाकलन m (samāklan)
• Kazakh: интегралды (ıyntegraldı), интегралдық (ıyntegraldıq)
• Manx: slane-earrooagh
• Portuguese: inteiro (pt)
• Romanian: integral (ro)
• Russian: целочи́сленный (ru) (celočíslennyj)
• Spanish: entero (es)
• Swedish: compounds with heltal (sv)
• Tagalog: buumbilangin
• Turkish: tümlevsel

math: relating to integration

• Arabic: تَكَامُلِيّ (ar) m (takāmuliyy)
• Bulgarian: интегрален (bg) (integralen)
• Esperanto: integrala
• Finnish: integraali- (fi)
• French: intégral (fr)
• Galician: integral (gl)
• Hindi: समाकलन m (samāklan)
• Kazakh: интегралды (ıyntegraldı), интегралдық (ıyntegraldıq)
• Spanish: integral (es)
• Turkish: tümlevsel
• Vietnamese: tích phân (vi)

The translations below need to be checked and inserted above into the appropriate translation tables. See instructions at Wiktionary:Entry layout § Translations.

Translations to be checked

• Hungarian: ^{(please verify)} integrál (hu) (2)
• Manx: ^{(please verify)} co-hymagh

Pronunciation 2

• (UK) enPR: ĭnʹtĭ-grəl, IPA^(key): /ˈɪntɪɡɹ(ə)l/
• (US) enPR: ĭnʹtə-grəl, IPA^(key): /ˈɪntəɡɹəl/, /ɪnˈtɛɡɹəl/

• Audio (UK):

  (file)

Noun

integral (plural integrals)

1. (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 limiting process on a sequence of partial sums. Denoted using a long s: ∫, or a variant thereof.

   The integral of a univariate real-valued function is the area under its curve; but be warned! Not all functions are integrable!

   1. (specifically) Any of several analytic formalizations of this operation: the Riemann integral, the Lebesgue integral, etc.
2. (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.

   The integral of $\frac{1}{x}$ on $[\frac{1}{2},1]$ is $\ln (2)$, but the integral of the same function on $(0,1]$ diverges. In notation, ${\int }_{\frac{1}{2}}^1\frac{1}{x}dx=\ln (2)$, but ${\int }_0^1\frac{1}{x}dx=\mathrm{\infty }$.

   Stokes' theorem relates the integral over a surface of the curl of a vector field to a line integral around the boundary of that surface: ${\int }_S\mathrm{\nabla }\times 𝐅\cdot d𝐒={\int }_{\partial S}𝐅\cdot d𝐫,$.

3. (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;

   Synonyms: antiderivative, indefinite integral, ∫

   Antonym: derivative

   The integral of $x^2$ is $\frac{x^3}{3}$ plus a constant.

   1. (mathematics, historical or obsolete) The fluent of a given fluxion in Newtonian calculus.

Derived terms

• Barnes integral
• Choquet integral
• contour integral
• definite integral
• double integral
• flux integral
• half-integral
• Hellinger integral
• improper integral
• indefinite integral
• integral calculus
• integral equation
• integral theory
• integral transform
• integration
• iterated integral
• Jackson integral
• Lebesgue integral
• line integral
• logarithmic integral
• Mellin-Barnes integral
• ordered integral domain
• particular integral
• path integral formalism
• Riemann integral
• Riemann-Stieltjes integral
• Sugeno integral
• surface integral
• triple integral
• Volterra integral equation
• volume integral

Related terms

• integer
• integrate

Translations

limit of sums

• Arabic: تَكَامُل (takāmul)
• Armenian: ինտեգրալ (hy) (integral)
• Belarusian: інтэгра́л m (intehrál)
• Bulgarian: интегра́л m (integrál)
• Catalan: integració (ca) f
• Chinese:

  Cantonese: 積分 / 积分 (zik¹ fan¹)

  Hakka: chit-fûn-ho̍k

  Mandarin: 積分 / 积分 (zh) (jī fēn)

• Czech: integrál (cs) m
• Dutch: integraal (nl)
• Esperanto: integralo (eo), integro
• Finnish: integraali (fi)
• French: intégrale (fr) f
• Galician: integral (gl) f
• Georgian: ინტეგრალი (inṭegrali)
• German: Integral (de) n
• Greek: ολοκλήρωμα (el) n (oloklíroma)
• Hebrew: אִינְטֶגְרָל (he) m (integrál), אַסְכֶּמֶת (he) f (askémet) (Academy of the Hebrew Language)
• Hungarian: integrál (hu)
• Icelandic: heildun f, tegrun f
• Italian: integrale (it) m
• Japanese: 積分 (ja) (せきぶん, sekibun)
• Kazakh: интеграл (ıyntegral)
• Khmer: អាំងតេក្រាល (km) (ʼangteekraal)
• Korean: 적분 (ko) (jeokbun)
• Kyrgyz: интеграл (integral)
• Malay: kamiran
• Manx: co-hym m
• Mongolian: интеграл (mn) (integral)
• Norwegian: integral n
• Polish: całka (pl) f, całka oznaczona (pl) f
• Portuguese: integral (pt) f
• Romanian: integrală (ro) f
• Russian: интегра́л (ru) m (intɛgrál)
• Serbo-Croatian:

  Cyrillic: интѐгра̄л m

  Latin: intègrāl (sh) m

• Spanish: integral (es) f
• Sundanese: integral
• Swedish: integral (sv) c
• Tagalog: lagom
• Thai: ปริพันธ์ (bpà-rí-pan)
• Turkish: integral (tr), tümlev (tr)
• Ukrainian: інтегра́л (uk) m (intehrál)
• Vietnamese: tích phân (vi)

antiderivative — see antiderivative

Anagrams

• Triangle, alerting, altering, relating, tanglier, teraglin, triangle

Albanian

Noun

integral m

1. (mathematics) integral (relating to integration)

Further reading

• Jani Thomai et al. (2006), “integral”, in FGJSH: Fjalor i gjuhës shqipe [Dictionary of the Albanian language] (in Albanian), Tiranë: Akademia e Shkencave e Shqipërisë
• FGJSSH: Fjalor i gjuhës së sotme shqipe [Dictionary of the modern Albanian language]‎^[2], 1980

Catalan

Etymology

Borrowed from Medieval Latin integrālis, from Latin integer.

Pronunciation

• IPA^(key): (Northern, Balearic, Central) [in.təˈɣɾal]
• IPA^(key): (Valencia, Northwestern) [in.teˈɣɾal]

• Audio (Barcelona):

  (file)

Adjective

integral m or f (masculine and feminine plural integrals)

1. integral
2. (of bread, etc.) wholegrain, wholemeal

Related terms

• enter
• integració
• integrar
• íntegre

Noun

integral f (plural integrals)

1. (mathematics) integral

Derived terms

• integral indefinida

Further reading

• “integral”, in Diccionari de la llengua catalana [Dictionary of the Catalan Language] (in Catalan), second edition, Institute of Catalan Studies [Catalan: Institut d'Estudis Catalans], April 2007

Indonesian

Etymology

Borrowed from Dutch integraal (“integral”), from French intégral, from Medieval Latin integrālis, from Latin integer (“entire”).

Pronunciation

• (Standard Indonesian) IPA^(key): /intəɡˈral/ [in̪.t̪ək̚ˈral]
• Rhymes: -al
• Syllabification: in‧teg‧ral

Adjective

intêgral (comparative lebih integral, superlative paling integral)

1. integral:
   1. constituting a whole together with other parts or factors; not omittable or removable

      Synonyms: bulat, utuh, sempurna

   2. (mathematics) relating to integration (“the process of finding the integral [noun] of a function”)

      Synonyms: kafi, rangkunan

Noun

intêgral (plural integral-integral)

1. (mathematics) integral

Related terms

• integrasi

Further reading

• “integral”, in Kamus Besar Bahasa Indonesia [Great Dictionary of the Indonesian Language] (in Indonesian), Jakarta: Agency for Language Development and Cultivation – Ministry of Education, Culture, Research, and Technology of the Republic of Indonesia, 2016

Middle French

Etymology

Borrowed from Medieval Latin integrālis, from Latin integer.

Adjective

integral m (feminine singular integrale, masculine plural integraux, feminine plural integrales)

1. integral, necessary to the function of the whole
2. whole; entire

Descendants

• French: intégral

References

• integral on Dictionnaire du Moyen Français (1330–1500) (in French)

Portuguese

Etymology

Etymology tree

Proto-Indo-European *né

Proto-Indo-European *n̥-

Proto-Indo-European *teh₂g-

Proto-Indo-European *-rós

Proto-Indo-European *n̥th₂gros

Proto-Italic *n̥tagros

Latin integer

Proto-Indo-European *h₂el-

Proto-Indo-European *-is

Proto-Indo-European *h₂élisder.?

Proto-Italic *-ālis

Latin -ālis

Medieval Latin integrālislbor.

Portuguese integral

Learned borrowing from Medieval Latin integrālis, from Latin integer + -ālis.

Pronunciation

• (Brazil) IPA^(key): /ĩ.teˈɡɾaw/ [ĩ.teˈɡɾaʊ̯]

 

• (Portugal) IPA^(key): /ĩ.tɨˈɡɾal/ [ĩ.tɨˈɣɾaɫ]

• (Portugal) IPA^(key): /ĩ.tɨˈɡɾal/ [ĩ.tɨˈɣɾaɫ]
  • (Southern Portugal) IPA^(key): /ĩ.tɨˈɡɾa.li/

• Rhymes: -al, -aw
• Hyphenation: in‧te‧gral

Adjective

integral m or f (plural integrais)

1. integral, whole, entire

   Synonyms: íntegro, inteiro, intacto, completo, total

2. whole (from which none of its constituents has been removed)

Derived terms

• integralmente

Noun

integral f or m (plural integrais)

2. (calculus) integral (limits of sums)

   Hyponyms: integral indefinida / integral indefinido, antiderivada, primitiva; integral definida / integral definido

   Coordinate term: derivada

Derived terms

• integração
• integral definida, integral definido
• integral indefinida, integral indefinido
• integralismo
  • neointegralismo
• integralista

Related terms

• integrar
  • integrado
  • integrante
  • integrável
• integridade
• íntegro

Further reading

• “integral”, in Dicionário Aulete Digital (in Portuguese), Rio de Janeiro: Lexikon Editora Digital, 2008–2026
• “integral”, in Dicionário Eletrônico Houaiss [Houaiss Electronic Dictionary] (in Portuguese), São Paulo: UOL, 2004–present
• “integral”, in Dicionário Infopédia da Língua Portuguesa (in Portuguese), Porto: Porto Editora, 2003–2026
• “integral”, in Michaelis Dicionário Brasileiro da Língua Portuguesa (in Portuguese), São Paulo: Editora Melhoramentos, 2015–2026, →ISBN
• “integral”, in Dicionário Priberam da Língua Portuguesa (in Portuguese), Lisbon: Priberam, 2008–2026

Romanian

Etymology

Borrowed from French intégral, Medieval Latin integrālis.

Pronunciation

• IPA^(key): /in.teˈɡral/

Adjective

integral m or n (feminine singular integrală, masculine plural integrali, feminine/neuter plural integrale)

1. integral

   Synonyms: întreg, complet

Declension

Declension of integral

		singular			plural
		masculine	neuter	feminine	masculine	neuter	feminine
nominative- accusative	indefinite	integral		integrală	integrali	integrale
	definite	integralul		integrala	integralii	integralele
genitive- dative	indefinite	integral		integrale	integrali	integrale
	definite	integralului		integralei	integralilor	integralelor

Related terms

• integru

Spanish

Etymology

Etymology tree

Proto-Indo-European *né

Proto-Indo-European *n̥-

Proto-Indo-European *teh₂g-

Proto-Indo-European *-rós

Proto-Indo-European *n̥th₂gros

Proto-Italic *n̥tagros

Latin integer

Proto-Indo-European *h₂el-

Proto-Indo-European *-is

Proto-Indo-European *h₂élisder.?

Proto-Italic *-ālis

Latin -ālis

Medieval Latin integrālislbor.

Spanish integral

Learned borrowing from Medieval Latin integrālis, from Latin integer + -ālis.

Pronunciation

• IPA^(key): /inteˈɡɾal/ [ĩn̪.t̪eˈɣ̞ɾal]

• Audio (Colombia):

  (file)

• Rhymes: -al
• Syllabification: in‧te‧gral

Adjective

integral m or f (masculine and feminine plural integrales)

1. integral
2. whole
3. brown (rice)
4. wholegrain

Derived terms

• arroz integral
• cálculo integral
• integralidad
• integralmente
• pan integral
• parte integral

Related terms

• íntegro

Noun

integral f (plural integrales)

1. (mathematics) integral

Further reading

• “integral”, in Diccionario de la lengua española [Dictionary of the Spanish Language] (in Spanish), 24th edition, Royal Spanish Academy [Spanish: Real Academia Española], 30 October 2026

Swedish

Etymology

Borrowed from Medieval Latin integrālis.

Pronunciation

• IPA^(key): /ɪntɛˈɡrɑːl/
• Hyphenation: in‧te‧gral
• Rhymes: -ɑːl

Noun

integral c

1. (mathematics) an integral

Declension

Declension of integral

		nominative	genitive
singular	indefinite	integral	integrals
	definite	integralen	integralens
plural	indefinite	integraler	integralers
	definite	integralerna	integralernas

Related terms

• integrand
• integration
• integrera

References

• “integral”, in Svensk ordbok [Dictionary of Swedish] (in Swedish)
• “integral”, in Svenska Akademiens ordlista [Wordlist of the Swedish Academy] (in Swedish)
• “integral”, in Svenska Akademiens ordbok [Dictionary of the Swedish Academy] (in Swedish)

Anagrams

• triangel

Tagalog

Etymology

Etymology tree

Proto-Indo-European *né

Proto-Indo-European *n̥-

Proto-Indo-European *teh₂g-

Proto-Indo-European *-rós

Proto-Indo-European *n̥th₂gros

Proto-Italic *n̥tagros

Latin integer

Proto-Indo-European *h₂el-

Proto-Indo-European *-is

Proto-Indo-European *h₂élisder.?

Proto-Italic *-ālis

Latin -ālis

Medieval Latin integrālislbor.

Spanish integralbor.

Tagalog integral

Borrowed from Spanish integral.

Pronunciation

• (Standard Tagalog) IPA^(key): /ʔinteɡˈɾal/ [ʔɪn̪.t̪ɛɡ̠ˈɾal]
• Rhymes: -al
• Syllabification: in‧teg‧ral

Adjective

integrál (Baybayin spelling ᜁᜈ᜔ᜆᜒᜄ᜔ᜇᜎ᜔)

1. integral

   Synonyms: buo, kompleto

Related terms

• integrante
• integrasyon

Turkish

Alternative forms

• entegral

Etymology

Borrowed from French intégral.

Pronunciation

• IPA^(key): /in.teɟˈɾal/, [in.tɛɟ.ˈɾalʲ]

Noun

integral (definite accusative integrali, plural integraller)

1. (mathematics) integral

   Synonym: tümlev

   ${\int }_a^{b}f(x)dx$

Declension

Declension of integral

singular plural nominative integral integraller definite accusative integrali integralleri dative integrale integrallere locative integralde integrallerde ablative integralden integrallerden genitive integralin integrallerin Possessive forms nominative singular plural 1st singular integralim integrallerim 2nd singular integralin integrallerin 3rd singular integrali integralleri 1st plural integralimiz integrallerimiz 2nd plural integraliniz integralleriniz 3rd plural integralleri integralleri definite accusative singular plural 1st singular integralimi integrallerimi 2nd singular integralini integrallerini 3rd singular integralini integrallerini 1st plural integralimizi integrallerimizi 2nd plural integralinizi integrallerinizi 3rd plural integrallerini integrallerini dative singular plural 1st singular integralime integrallerime 2nd singular integraline integrallerine 3rd singular integraline integrallerine 1st plural integralimize integrallerimize 2nd plural integralinize integrallerinize 3rd plural integrallerine integrallerine locative singular plural 1st singular integralimde integrallerimde 2nd singular integralinde integrallerinde 3rd singular integralinde integrallerinde 1st plural integralimizde integrallerimizde 2nd plural integralinizde integrallerinizde 3rd plural integrallerinde integrallerinde ablative singular plural 1st singular integralimden integrallerimden 2nd singular integralinden integrallerinden 3rd singular integralinden integrallerinden 1st plural integralimizden integrallerimizden 2nd plural integralinizden integrallerinizden 3rd plural integrallerinden integrallerinden genitive singular plural 1st singular integralimin integrallerimin 2nd singular integralinin integrallerinin 3rd singular integralinin integrallerinin 1st plural integralimizin integrallerimizin 2nd plural integralinizin integrallerinizin 3rd plural integrallerinin integrallerinin

Further reading

• “integral”, in Turkish dictionaries, Türk Dil Kurumu
• Nişanyan, Sevan (2002–), “integral”, in Nişanyan Sözlük
• Ayverdi, İlhan (2010), “integral”, in Misalli Büyük Türkçe Sözlük, a reviewed and expanded single-volume edition, Istanbul: Kubbealtı Neşriyatı