integral — English Dictionary

Adjective

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

   Synonyms

   immanentinherentnecessary

   Examples

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

   More examples (1)

   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. Of, pertaining to, or being an integer.

3. Relating to integration (“the process of finding the integral [noun] of a function”).

4. Being the root of some monic polynomial in A.

   Usage

   • of a ring element in a ring B relative to a subring A

5. Whole; undamaged.

   Usage

   • Obsolete

   Examples

   A local motion keepeth bodies integral.

Noun

1. 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.

   Examples

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

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

   Usage

   • specifically

3. 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.

   Examples

   More examples (2)

   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;,.

4. 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

   antiderivativeindefinite integralmul:∫

   Antonyms

   derivative

   Examples

   The integral of x² is #92;frac#123;x³#125;#123;3#125; plus a constant.

5. 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 fluent of a given fluxion in Newtonian calculus.

   Usage

   • Historical
   • Obsolete

Adjective

1. existing as an essential constituent or characteristic

   Synonyms

   built-inconstitutionalinbuiltinherent

   Examples

   the Ptolemaic system with its built-in concept of periodicity

   a constitutional inability to tell the truth

2. constituting the undiminished entirety; lacking nothing essential especially not damaged

   Synonyms

   entireintact

   Examples

   was able to keep the collection entire during his lifetime

   fought to keep the union intact

3. of or denoted by an integer

Noun

1. the result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x)

   Broader terms

   calculationcomputationfiguringreckoning