integral

comparative more integral · superlative most integral

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

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

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

   1 more example

   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.

   2021 December 17, Vanessa Friedman, “The Verdict on the Elizabeth Holmes Trial Makeover”, in The New York Times, →ISSN, archived from the original on 12 Apr 2022

   Synonyms: immanent, inherent, necessary, built-in, essential, immanent, inborn, resident

2. mathematicsOf, pertaining to, or being an integer.
3. mathematicsRelating to integration (“the process of finding the integral [noun] of a function”).
4. commutative algebraalgebraBeing the root of some monic polynomial in A.
5. obsoleteWhole; undamaged.

   A local motion keepeth bodies integral.

   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

plural integrals

1. mathematicsOne 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. specificallymathematicsAny of several analytic formalizations of this operation: the Riemann integral, the Lebesgue integral, etc.
2. mathematicsA 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.

   2 more examples

   The integral of #92;frac#123;1#125;#123;x#125; on #91;#92;frac#123;1#125;#123;2#125;,1#93; is #92;ln(2), but the integral of the same function on (0,1#93; diverges. In notation, #92;int#95;#92;frac#123;1#125;#123;2#125;¹#92;frac#123;1#125;#123;x#125;dx#61;#92;ln(2), but #92;int#95;0¹#92;frac#123;1#125;#123;x#125;dx#61;#92;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: #92;int#95;S#92;nabla#92;times#92;mathbf#123;F#125;#92;cdotd#92;mathbf#123;S#125;#61;#92;int#95;#123;#92;partialS#125;#92;mathbf#123;F#125;#92;cdotd#92;mathbf#123;r#125;,.

3. mathematicsAn 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 x² is #92;frac#123;x³#125;#123;3#125; plus a constant.

   Synonyms: antiderivative, indefinite integral, mul:∫ · Antonyms: derivative

   1. historicalobsoletemathematicsThe fluent of a given fluxion in Newtonian calculus.

      Synonyms: antiderivative, indefinite integral, mul:∫ · Antonyms: derivative

1. abbreviationspaceESAAbbreviation of International Gamma-Ray Astrophysics Laboratory (a satellite of the European Space Agency).