Antiderivative vs. Integral — What's the Difference?

Focuses on function relationship.

Includes both definite and indefinite forms.

Inverse of differentiation.

Applied in various fields.

Essential in calculus.

Calculates areas and accumulations.

Basis for indefinite integrals.

Indispensable in mathematical analysis.

Includes the constant of integration C.

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration.

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f.

Necessary to make a whole complete; essential or fundamental

See indefinite integral.

Of or denoted by an integer.

(calculus) A function whose derivative is a given function; an indefinite integral Category:en:Functions

A function of which a given function is the derivative, i.e. which yields that function when differentiated, and which may express the area under the curve of a graph of the function.

Essential or necessary for completeness; constituent

Possessing everything essential; entire.

Expressed or expressible as or in terms of integers.

Expressed as or involving integrals.

A complete unit; a whole.

A number computed by a limiting process in which the domain of a function, often an interval or planar region, is divided into arbitrarily small units, the value of the function at a point in each unit is multiplied by the linear or areal measurement of that unit, and all such products are summed.

A definite integral.

An indefinite integral.

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

(mathematics) Of, pertaining to, or being an integer.

(mathematics) Relating to integration.

(obsolete) Whole; undamaged.

(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.

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

(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.

(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 fluent of a given fluxion in Newtonian calculus.

Lacking nothing of completeness; complete; perfect; uninjured; whole; entire.

Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.

Of, pertaining to, or being, a whole number or undivided quantity; not fractional.

A whole; an entire thing; a whole number; an individual.

An expression which, being differentiated, will produce a given differential. See differential Differential, and Integration. Cf. Fluent.

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

Existing as an essential constituent or characteristic;

Constituting the undiminished entirety; lacking nothing essential especially not damaged;

The constant C accounts for the fact that there are infinitely many functions that can serve as antiderivatives, differing only by a constant.

An antiderivative is used to find a function whose derivative is the given function, essential in solving differential equations and understanding function behavior.

An integral, encompassing both definite and indefinite forms, calculates areas under curves and accumulations, whereas an antiderivative focuses on reversing the process of differentiation.

Definite integrals are significant for calculating precise quantities like areas, volumes, and other accumulations, crucial in various scientific and engineering applications.

Finding an antiderivative involves knowing the basic derivatives and applying the rules of integration, often requiring manipulation of the function to match a known form.

Most elementary functions have antiderivatives, but some functions, especially those not continuous everywhere, may not have antiderivatives expressible in terms of elementary functions.

Integrals are used to model and solve problems related to rates of change, area and volume calculations, economics, physics problems like motion and energy, and engineering design.

While integration is primarily for continuous functions, discrete analogs, like summations, serve a similar purpose in accumulating quantities over discrete intervals.

Yes, antiderivatives and indefinite integrals are essentially the same, both representing the general solution to the process of differentiation.

A practical example includes calculating the total distance traveled by an object when given its velocity over time, using the definite integral to sum up all infinitesimal distances.