• (British) enPR: ĭnʹtĭ-grəl, IPA: /ˈɪntɪɡɹ(ə)l/
• (America) enPR: ĭnʹtə-grəl, IPA: /ˈɪntəɡɹəl/, /ɪnˈtɛɡɹəl/
• (British) IPA: /ˈɪntɪɡr(ə)l/, /ɪnˈtɛɡr(ə)l/
• (America) enPR: ĭn-tĕgʹrəl, IPA: /ˈɪntəɡɹəl/, /ɪnˈtɛɡɹəl/

integral

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

   Synonyms: immanent, inherent, necessary, 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.

2. (mathematics) Of, pertaining to, or being an integer.
3. (mathematics) Relating to integration.
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.

• German: integral, ein Ganzes ausmachend, zur Erhaltung des Ganzen unentbehrlich (descriptive)
• Portuguese: integral
• Russian: неотъе́млемый

• French: intégral
• German: ganzzahlig
• Portuguese: inteiro
• Russian: целочи́сленный
• Spanish: entero

• French: intégral
• Spanish: integral

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 = \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 \nabla\times \mathbf{F}\cdot d\mathbf{S} = \int_{\partial S} \mathbf{F}\cdot d\mathbf{r},.

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;

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

   Synonyms: antiderivative, indefinite integral, ∫

   Antonyms: derivative

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

• French: intégrale
• German: Integral
• Italian: integrale
• Portuguese: integral
• Russian: интегра́л
• Spanish: integral

1. (space, ESA) Abbreviation of International Gamma-Ray Astrophysics Laboratory a satellite of the European Space Agency