NationMaster - Encyclopedia: Antiderivative

In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i.e., F ′ = f. The process of solving for antiderivatives is antidifferentiation (or indefinite integration). They are also called integrals, but this usage is not universally accepted. Antiderivatives are related to definite integrals through the fundamental theorem of calculus, and provide a convenient means for calculating the integrals of many functions. Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ... Partial plot of a function f. ... In mathematics, a derivative is the rate of change of a quantity. ... In calculus, the integral of a function is an extension of the concept of a sum. ... In calculus, the integral of a function is an extension of the concept of a sum. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ...

1 Example
2 Uses and properties
3 Techniques of integration
4 Antiderivatives of non-continuous functions
- 4.1 Some examples
5 See also
6 References

Example

The function F(x) = x³/3 is an antiderivative of f(x) = x². As the derivative of a constant is zero, x² will have an infinite number of antiderivatives; such as (x³/3) + 0, (x³ / 3) + 7, (x³ / 3) − 36, etc. Thus, the antiderivative family of x² is collectively referred to by F(x) = (x³ / 3) + C; where C is an arbitrary constant known as the constant of integration. Essentially, the graphs of antiderivatives of a given function are vertical translations of each other; each graph's location depending upon the value of C. In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... The infinity symbol âˆž in several typefaces. ... In mathematics, an index set is another name for a function domain. ... In calculus, the indefinite integral of a given function (i. ... In function graphing, a vertical translation is a related graph which, for every point (x, y); has a y value which differs from another graph, by exactly some constant c. ... Value in mathematics refers to the quantity that is represented by a variable. ...

Uses and properties

Antiderivatives are important because they can be used to compute integrals, using the fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then: In calculus, the integral of a function is an extension of the concept of a sum. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ...

$intlimits _a^b f(x), dx = F(b) - F(a).$

Because of this, the set of all antiderivatives of a given function f is sometimes called the general integral or indefinite integral of f and is written using the integral symbol with no bounds: In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...

$int f(x), dx.$

It is critical to remember that an integral is not the same, in general, as the means for evaluating it; and the function that an integral implies stands apart from that means - in the case of single-variable integrals, from antiderivatives.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G(x) = F(x) + C for all x. C is called the arbitrary constant of integration. If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals. For instance In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In calculus, the indefinite integral of a given function (i. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...

$F(x)=begin{cases}-frac{1}{x}+C_1quad x<0-frac{1}{x}+C_2quad x>0end{cases}$

is the most general antiderivative of $f (x) = 1 / x 2$ on its natural domain $(-infty,0)cup(0,infty).$

Every continuous function f has an antiderivative, and one antiderivative F is given by the integral of f with variable upper boundary: In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

$F(x) = intlimits_a^x f(t),dt.$

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 the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ...

There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions (like polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations). Examples of these are In mathematics, several functions are important enough to deserve their own name. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... The exponential function is one of the most important functions in mathematics. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: - Cosine, - Sine, - Tangent, - Cosecant, - Secant, - Cotangent In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among... In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ...

$int e^{-x^2},dx,qquad int frac{sin(x)}{x},dx,qquad intfrac{1}{ln x},dx.$

See also differential Galois theory for a more detailed discussion. // Motivation and basic idea In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. ...

Techniques of integration

Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives. For some elementary functions, it is impossible to find an antiderivative in terms of other elementary functions. See the article on elementary functions for further information. In differential algebra, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ − × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by...

We have various methods at our disposal:

the linearity of integration allows us to break complicated integrals into simpler ones
integration by substitution, often combined with trigonometric identities or the natural logarithm
integration by parts to integrate products of functions
the inverse chain rule method, a special case of integration by substitution
the method of partial fractions in integration allows us to integrate all rational functions (fractions of two polynomials)
the Risch algorithm
integrals can also be looked up in a table of integrals
when integrating multiple times, we can use certain additional techniques, see for instance double integrals and polar coordinates, the Jacobian and the Stokes theorem
computer algebra systems 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
if a function has no elementary antiderivative (for instance, exp(x²)), its integral can be approximated using numerical integration

In calculus, linearity is a fundamental property of the integral that follows from the sum rule in integration and the constant factor rule in integration. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... In calculus, the inverse chain rule is a method of integrating a function which relies on guessing the integral of that function, and then differentiating back using the chain rule. ... In integral calculus, the use of partial fractions is required to integrate the general rational function. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... The Risch algorithm is an algorithm for the calculus operation of indefinite integration (i. ... It has been suggested that this article or section be merged with List of integrals. ... In mathematical analysis, there is a serious distinction between a double integral and an iterated integral. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... Stokes Theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ...

Antiderivatives of non-continuous functions

To illustrate some of the subtleties of the fundamental theorem of calculus, it is instructive to consider what kinds of non-continuous functions might have antiderivatives. While there are still open questions in this area, it is known that: The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ...

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.

We first state some general facts and then provide some illustrative examples. Throughout, we assume that the domains of our functions are open intervals. If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...

A necessary, but not sufficient, condition for a function f to have an antiderivative is that f have the intermediate value property. That is, if [a,b] is a subinterval of the domain of f and d is any real number between f(a) and f(b), then f(c)=d for some c between a and b. To see this, let F be an antiderivative of f and consider the continuous function g(x)=F(x)-dx on the closed interval [a, b]. Then g must have either a maximum or minimum c in the open interval (a,b) and so 0=g′(c)=f(c)-d.
The set of discontinuities of f 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 f having an antiderivative, which has the given set as its set of discontinuities.
If f has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of Lebesgue measure 0, then an antiderivative may be found by integration.
If f has an antiderivative F on a closed interval [a,b], then for any choice of partition $a=x_0<x_1<x_2<dots<x_n=b$ , if one chooses sample points $x_i^*in[x_{i-1},x_i]$ as specified by the mean value theorem, then the corresponding Riemann sum telescopes to the value F(b)-F(a).

$sum_{i=1}^n f(x_i^*)(x_i-x_{i-1}) = sum_{i=1}^n [F(x_i)-F(x_{i-1})] = F(x_n)-F(x_0) = F(b)-F(a)$

However if the set of discontinuities of f has positive Lebesgue measure, a different choice of sample points $x_i^*$ will give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below.

In analysis, the intermediate value theorem is either of two theorems of which an account is given below. ... In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ... In mathematics, an FÏƒ set (said F-sigma set) is a countable union of closed sets. ... The term bounded appears in different parts of mathematics where a notion of size can be given. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ... In mathematics, telescoping series is an informal expression referring to a series whose sum can be found by exploiting the circumstance that nearly every term cancels with a succeeding or preceding term. ...

Some examples

The function

$f(x)=2xsinleft(frac{1}{x}right)-cosleft(frac{1}{x}right)$

with $fleft(0right)=0$ is not continuous at $x = 0$ but has the antiderivative

$Fleft(xright)=x^2sinleft(frac{1}{x}right)$

with $Fleft(0right)=0$ . Since f is bounded on closed finite intervals and is only discontinuous at 0, the antiderivative F may be obtained by integration: $F(x)=int_0^x f(t),dt$ .
The function

$f(x)=2xsinleft(frac{1}{x^2}right)-frac{2}{x}cosleft(frac{1}{x^2}right)$

with $fleft(0right)=0$ is not continuous at $x = 0$ but has the antiderivative

$F(x)=x^2sinleft(frac{1}{x^2}right)$

with $Fleft(0right)=0$ . Unlike Example 1, f(x) is unbounded in any interval containing 0, so the Riemann integral is undefined.
If f(x) is the function in Example 1 and F is its antiderivative, and ${x_n}_{nge1}$ is a dense countable subset of the open interval $left(-1,1right)$ , then the function

$g(x)=sum_{n=1}^infty frac{f(x-x_n)}{2^n}$

has as antiderivative In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of... In mathematics the term countable set is used to describe the size of a set, e. ...

$G(x)=sum_{n=1}^infty frac{F(x-x_n)}{2^n}.$

The set of discontinuities of g is precisely the set ${x_n}_{nge1}$ . Since g is bounded on closed finite intervals and the set of discontinuities has measure 0, the antiderivative G may be found by integration.
Let ${x_n}_{nge1}$ be a dense countable subset of the open interval $left(-1,1right)$ . Consider the everywhere continuous strictly increasing function

$F(x)=sum_{n=1}^inftyfrac{1}{2^n}(x-x_n)^{1/3}.$

It can be shown that In mathematics, the term dense has at least three different meanings. ... In mathematics the term countable set is used to describe the size of a set, e. ...

$F'(x)=sum_{n=1}^inftyfrac{1}{3cdot2^n}(x-x_n)^{-2/3}$

Figure 1.

Figure 2.

for all values x where the series converges, and that the graph of F(x) has vertical tangent lines at all other values of x. In particular the graph has vertical tangent lines at all points in the set ${x_n}_{nge1}$ . Image File history File links Antideriv1. ... Image File history File links Antideriv1. ... Image File history File links Antideriv2. ... Image File history File links Antideriv2. ...

Moreover $Fleft(xright)ge0$ for all x where the derivative is defined. It follows that the inverse function $G = F - 1$ is differentiable everywhere and that

$gleft(xright)=G'left(xright)=0$

for all x in the set ${F(x_n)}_{nge1}$ which is dense in the interval $left[Fleft(-1right),Fleft(1right)right]$ . Thus g has an antiderivative G. On the other hand, it can not be true that

$int_{F(-1)}^{F(1)}g(x),dx=GF(1)-GF(-1)=2,$

since for any partition of $left[Fleft(-1right),Fleft(1right)right]$ , one can choose sample points for the Riemann sum from the set ${F(x_n)}_{nge1}$ , giving a value of 0 for the sum. It follows that g has a set of discontinuities of positive Lebesgue measure. Figure 1 on the right shows an approximation to the graph of g(x) where ${x_n=cos(n)}_{nge1}$ and the series is truncated to 8 terms. Figure 2 shows the graph of an approximation to the antiderivative G(x), also truncated to 8 terms. On the other hand if the Riemann integral is replaced by the Lebesgue integral, then Fatou's lemma or the dominated convergence theorem shows that g does satisfy the fundamental theorem of calculus in that context. In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... Fatous lemma establishes an inequality relating the integral (in the sense of Lebesgue) of the limit inferior of a sequence of functions to the limit inferior of the sequence of integrals of the functions. ... In mathematics, Lebesgues dominated convergence theorem states that if a sequence { fn : n = 1, 2, 3, ... } of real-valued measurable functions on a measure space S converges almost everywhere, and is dominated (explained below) by some nonnegative function g in , then To say that the sequence is dominated by...
In Examples 3 and 4, the sets of discontinuities of the functions g are dense only in a finite open interval $left(a,bright)$ . However these examples can be easily modified so as to have sets of discontinuities which are dense on the entire real line $(-infty,infty)$ . Let

Then $gleft(lambda(x)right)lambda'(x)$ has a dense set of discontinuities on $(-infty,infty)$ and has antiderivative $Gcdotlambda.$
Using a similar method as in Example 5, one can modify g in Example 4 so as to vanish at all rational numbers. If one uses a naive version of the Riemann integral defined as the limit of left-hand or right-hand Riemann sums over regular partitions, one will obtain that the integral of such a function g over an interval $left[a,bright]$ is 0 whenever a and b are both rational, instead of $Gleft(bright)-Gleft(aright)$ . Thus the fundamental theorem of calculus will fail spectacularly.

In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...

References

Introduction to Classical Real Analysis, by Karl R. Stromberg; Wadsworth, 1981 (see also)
Historical Essay On Continuity Of Derivatives, by Dave L. Renfro; http://groups.google.com/group/sci.math/msg/814be41b1ea8c024

Category: Integral calculus

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