
In
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
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 ...
whose
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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 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 the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
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 scientific study of matter, its Elementary particle, 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 whi ...
, antiderivatives arise in the context of
rectilinear motion (e.g., in explaining the relationship between
position,
velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
and
acceleration
In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
).
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, ...
equivalent of the notion of antiderivative is
antidifference.
Examples
The function
is an antiderivative of
, since the derivative of
is
. Since the derivative of a
constant is
zero
0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
,
will have an
infinite number of antiderivatives, such as
, etc. Thus, all the antiderivatives of
can be obtained by changing the value of in
, where is an arbitrary constant known as the
constant of integration. The
graphs of antiderivatives of a given function are
vertical translations of each other, with each graph's vertical location depending upon the
value .
More generally, the
power function has antiderivative
if , and
if .
In
physics
Physics is the scientific study of matter, its Elementary particle, 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 whi ...
, the integration of
acceleration
In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
yields
velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
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 ...
:
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 derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, inte ...
: if is an antiderivative of the continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
over the interval disjoint union
In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...
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
-\dfrac + c_1 & x<0 \\ ex-\dfrac + c_2 & 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 small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
has an antiderivative, and one antiderivative is given by the definite integral of with variable upper boundary:
F(x) = \int_a^x f(t)\,\mathrmt ~,
for any in the domain of . 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 derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, inte ...
.
There are many 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, a ...
s whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions. Elementary functions are polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s, exponential functions, logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
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 ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted Domain of a functi ...
and their combinations under composition and linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
. Examples of these nonelementary integrals are
* the error function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as:
\operatorname z = \frac\int_0^z e^\,\mathrm dt.
The integral here is a complex Contour integrat ...
\int e^\,\mathrmx,
* the Fresnel function \int \sin x^2\,\mathrmx,
* the sine integral
In mathematics, trigonometric integrals are a indexed family, family of nonelementary integrals involving trigonometric functions.
Sine integral
The different sine integral definitions are
\operatorname(x) = \int_0^x\frac\,dt
\operato ...
\int \frac\,\mathrmx,
* 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 theory, number theoretic significance. In particular, according to the prime number the ...
\int\frac\,\mathrmx, and
* sophomore's dream \int x^\,\mathrmx.
For a more detailed discussion, see also Differential Galois theory.
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 and nonelementary integral.
There exist many properties and techniques for finding antiderivatives. These include, among others:
* The linearity of integration (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 c ...
, often combined with trigonometric identities or the natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
* The inverse chain rule method (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 antiderivati ...
(to integrate products of functions)
* Inverse function integration (a formula that expresses the antiderivative of the inverse of an invertible and continuous function , in terms of and the antiderivative of ).
* The method of partial fractions in integration (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 ...
s—fractions of two polynomials)
* The Risch algorithm
* Additional techniques for multiple integrations (see for instance double integrals, polar coordinates
In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are
*the point's distance from a reference ...
, the Jacobian and the Stokes' theorem)
* Numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
(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 (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.
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, 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.
* The set of discontinuities of must be a meagre set. 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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
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, 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's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...
, then the corresponding Riemann sum telescopes
A telescope is a device used to observe distant objects by their emission, Absorption (electromagnetic radiation), absorption, or Reflection (physics), reflection of electromagnetic radiation. Originally, it was an optical instrument using len ...
to the value F(b)-F(a). \begin
\sum_^n f(x_i^*)(x_i-x_) & = \sum_^n (x_i)-F(x_)\\
& = 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 \mathrm^ \mathrmx = \mathrm^ + C
* \int a^ \mathrmx = \frac + C;\ a > 0,\ a \neq 1
* \int \frac\sqrt\ \mathrmx = \arcsin\left(\frac\right) + C
* \int \frac\ \mathrmx = \frac\arctan\left(\frac\right) + C
See also
* Antiderivative (complex analysis)
* Formal antiderivative
* Jackson integral
* Lists of integrals
* 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 formula for a differentiable function ''F''(''x'') such that
:\frac = f(x ...
* Area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
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
Function Calculator
from WIMS
at HyperPhysics
Antiderivatives and indefinite integrals
at the Khan Academy
Integral calculator
at Symbolab
The Antiderivative
at MIT
Introduction to Integrals
at SparkNotes
Antiderivatives
at Harvy Mudd College
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Integral calculus
Linear operators in calculus