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In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
is a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
whose
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
is equal to the original function . This can be stated symbolically as . The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called ''differentiation'', which is the process of finding a derivative. Antiderivatives are often denoted by capital
Roman letters The Latin script, also known as Roman script, is an alphabetic writing system based on the letters of the classical Latin alphabet, derived from a form of the Greek alphabet which was in use in the ancient Greece, Greek city of Cumae, in southe ...
such as and . Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval. In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, antiderivatives arise in the context of
rectilinear motion Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...
(e.g., in explaining the relationship between
position Position often refers to: * Position (geometry), the spatial location (rather than orientation) of an entity * Position, a job or occupation Position may also refer to: Games and recreation * Position (poker), location relative to the dealer * ...
,
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
and
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
). The
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...
equivalent of the notion of antiderivative is
antidifference In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by \sum _x or \Delta^ , is the linear operator, inverse of the forward difference operator \Delta . It relates to the forward difference operator ...
.


Examples

The function F(x) = \tfrac is an antiderivative of f(x) = x^2, since the derivative of \tfrac is x^2, and since the derivative of a constant is
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
, x^2 will have an
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music * Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
number of antiderivatives, such as \tfrac, \tfrac+1, \tfrac-2, etc. Thus, all the antiderivatives of x^2 can be obtained by changing the value of in F(x) = \tfrac+c, where is an arbitrary constant known as the
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected ...
. Essentially, the
graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of antiderivatives of a given function are
vertical translation In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every ...
s of each other, with each graph's vertical location depending upon the
value Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...
. More generally, the
power function Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
f(x) = x^n has antiderivative F(x) = \tfrac + c if , and F(x) = \ln , x, + c if . In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the integration of
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
yields
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
plus a constant. The constant is the initial velocity term that would be lost upon taking the derivative of velocity, because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion (position, velocity, acceleration, and so on). Thus, integration produces the relations of acceleration, velocity and
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
: :\int a\ \mathrmt = v + C :\int v\ \mathrmt = d + C


Uses and properties

Antiderivatives can be used to compute definite integrals, using the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
: if is an antiderivative of the
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
function over the interval ,b/math>, then: :\int_a^b f(x)\,\mathrmx = F(b) - F(a). Because of this, each of the infinitely many antiderivatives of a given function may be called the "indefinite integral" of ''f'' and written using the integral symbol with no bounds: :\int f(x)\,\mathrmx. If is an antiderivative of , and the function is defined on some interval, then every other antiderivative of differs from by a constant: there exists a number such that G(x) = F(x)+c for all . is called the
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected ...
. If the domain of is a
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...
of two or more (open) intervals, then a different constant of integration may be chosen for each of the intervals. For instance :F(x)=\begin-\frac+c_1\quad x<0\\-\frac+c_2\quad x>0\end is the most general antiderivative of f(x)=1/x^2 on its natural domain (-\infty,0)\cup(0,\infty). Every
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
has an antiderivative, and one antiderivative is given by the definite integral of with variable upper boundary: :F(x)=\int_0^x f(t)\,\mathrmt. Varying the lower boundary produces other antiderivatives (but not necessarily all possible antiderivatives). This is another formulation of the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
. There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
s (like
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s,
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
s,
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s,
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
,
inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...
and their combinations). Examples of these are :\int e^\,\mathrmx,\qquad \int \sin x^2\,\mathrmx, \qquad\int \frac\,\mathrmx,\qquad \int\frac\,\mathrmx,\qquad \int x^\,\mathrmx. ''From left to right, the functions are the
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non-elementary ...
, the
Fresnel function 250px, Plots of and . The maximum of is about . If the integrands of and were defined using instead of , then the image would be scaled vertically and horizontally (see below). The Fresnel integrals and are two transcendental functions n ...
, the
sine integral In mathematics, trigonometric integrals are a indexed family, family of integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operatorname(x) = -\int ...
, the
logarithmic integral function In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
and
sophomore's dream In mathematics, the sophomore's dream is the pair of identities (especially the first) :\begin \int_0^1 x^\,dx &= \sum_^\infty n^ \\ \end :\begin \int_0^1 x^x \,dx &= \sum_^\infty (-1)^n^ = - \sum_^\infty (-n)^ \end discovered in 1697 by Jo ...
.'' For a more detailed discussion, see also
Differential Galois theory In mathematics, differential Galois theory studies the Galois groups of differential equations. Overview Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential field ...
.


Techniques of integration

Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives (indeed, there is no pre-defined method for computing indefinite integrals). For some elementary functions, it is impossible to find an antiderivative in terms of other elementary functions. To learn more, see
elementary functions In mathematics, an elementary function is a function (mathematics), function of a single variable (mathematics), variable (typically Function of a real variable, real or Complex analysis#Complex functions, complex) that is defined as taking addit ...
and
nonelementary integral In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an ''elementary function'' (i.e. a function constructed from a finite number of quotients of constan ...
. There exist many properties and techniques for finding antiderivatives. These include, among others: * The
linearity of integration 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. Along with d ...
(which breaks complicated integrals into simpler ones) *
Integration by substitution In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can ...
, often combined with
trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
or the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
* The
inverse chain rule method In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can ...
(a special case of integration by substitution) *
Integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
(to integrate products of functions) *
Inverse function integration In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse f^ of a continuous and invertible function in terms of f^ and an antiderivative of This formula was publishe ...
(a formula that expresses the antiderivative of the inverse of an invertible and continuous function , in terms of the antiderivative of and of ). * The method of
partial fractions in integration In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
(which allows us to integrate all
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s—fractions of two polynomials) * The
Risch algorithm In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra ...
* Additional techniques for multiple integrations (see for instance
double integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
s,
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
, the Jacobian and the
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
) *
Numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
(a technique for approximating a definite integral when no elementary antiderivative exists, as in the case of ) * Algebraic manipulation of integrand (so that other integration techniques, such as integration by substitution, may be used) *
Cauchy formula for repeated integration The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress ''n'' antidifferentiations of a function into a single integral (cf. Cauchy's formula). Scalar case Let ''f'' be a continuous function on the ...
(to calculate the -times antiderivative of a function) \int_^x \int_^ \cdots \int_^ f(x_n) \,\mathrmx_n \cdots \, \mathrmx_2\, \mathrmx_1 = \int_^x f(t) \frac\,\mathrmt.
Computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s can be used to automate some or all of the work involved in the symbolic techniques above, which is particularly useful when the algebraic manipulations involved are very complex or lengthy. Integrals which have already been derived can be looked up in a
table of integrals Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not ...
.


Of non-continuous functions

Non-continuous functions can have antiderivatives. While there are still open questions in this area, it is known that: * Some highly pathological functions with large sets of discontinuities may nevertheless have antiderivatives. * In some cases, the antiderivatives of such pathological functions may be found by
Riemann integration In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
, while in other cases these functions are not Riemann integrable. Assuming that the domains of the functions are open intervals: * A necessary, but not sufficient, condition for a function to have an antiderivative is that have the intermediate value property. That is, if is a subinterval of the domain of and is any real number between and , then there exists a between and such that . This is a consequence of
Darboux's theorem Darboux's theorem is a theorem in the mathematics, mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the ...
. * The set of discontinuities of must be a
meagre set In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
. This set must also be an F-sigma set (since the set of discontinuities of any function must be of this type). Moreover, for any meagre F-sigma set, one can construct some function having an antiderivative, which has the given set as its set of discontinuities. * If has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
0, then an antiderivative may be found by integration in the sense of Lebesgue. In fact, using more powerful integrals like the
Henstock–Kurzweil integral In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide ...
, every function for which an antiderivative exists is integrable, and its general integral coincides with its antiderivative. * If has an antiderivative on a closed interval ,b/math>, then for any choice of partition a=x_0 if one chooses sample points x_i^*\in _,x_i/math> as specified by the
mean value theorem In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
, then the corresponding Riemann sum
telescopes A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to observe ...
to the value F(b)-F(a). ::\begin \sum_^n f(x_i^*)(x_i-x_) & = \sum_^n
(x_i)-F(x_) X, or x, is the twenty-fourth and third-to-last Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its English a ...
\\ & = F(x_n)-F(x_0) = F(b)-F(a) \end :However if is unbounded, or if is bounded but the set of discontinuities of has positive Lebesgue measure, a different choice of sample points x_i^* may give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below.


Some examples


Basic formulae

* If f(x) = g(x), then \int g(x) \mathrmx = f(x) + C. * \int 1\ \mathrmx = x + C * \int a\ \mathrmx = ax + C * \int x^n \mathrmx = \frac + C;\ n \neq -1 * \int \sin\ \mathrmx = -\cos + C * \int \cos\ \mathrmx = \sin + C * \int \sec^2\ \mathrmx = \tan + C * \int \csc^2\ \mathrmx = -\cot + C * \int \sec\tan\ \mathrmx = \sec + C * \int \csc\cot\ \mathrmx = -\csc + C * \int \frac\ \mathrmx = \ln, x, + C * \int e^ \mathrmx = e^ + C * \int a^ \mathrmx = \frac + C;\ a > 0,\ a \neq 1


See also

*
Antiderivative (complex analysis) In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function ''g'' is a function whose complex derivative is ''g''. More precisely, given an open set U in the complex plane and a function g:U\to \ ...
* Formal antiderivative *
Jackson integral In q-analog theory, the Jackson integral series in the theory of special functions that expresses the operation inverse to q-differentiation. The Jackson integral was introduced by Frank Hilton Jackson. For methods of numerical evaluation, see ...
*
Lists of integrals Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does no ...
*
Symbolic integration In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or ''indefinite integral'', of a given function ''f''(''x''), i.e. to find a differentiable function ''F''(''x'') such that :\frac = f(x). This is also ...
*
Area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...


Notes


References


Further reading

* ''Introduction to Classical Real Analysis'', by Karl R. Stromberg; Wadsworth, 1981 (se
also

''Historical Essay On Continuity Of Derivatives''
by Dave L. Renfro


External links


Wolfram Integrator
— Free online symbolic integration with
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...

Mathematical Assistant on Web
— symbolic computations online. Allows users to integrate in small steps (with hints for next step (integration by parts, substitution, partial fractions, application of formulas and others), powered by Maxima
Function Calculator
from WIMS

at
HyperPhysics ''HyperPhysics'' is an educational website about physics topics. The information architecture of the website is based on HyperCard, the platform on which the material was originally developed, and a thesaurus organization, with thousands of contr ...

Antiderivatives and indefinite integrals
at the
Khan Academy Khan Academy is an American non-profit educational organization created in 2008 by Sal Khan. Its goal is creating a set of online tools that help educate students. The organization produces short lessons in the form of videos. Its website also in ...

Integral calculator
at
Symbolab Symbolab is an answer engine that provides step-by-step solutions to mathematical problems in a range of subjects. It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020 ...

The Antiderivative
at
MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...

Introduction to Integrals
at SparkNotes
Antiderivatives
at Harvy Mudd College {{Authority control Integral calculus Linear operators in calculus