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| {{for|lists of antiderivatives or primitive functions|lists of integrals}}
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| [[File:Slope Field.png|thumb|The [[slope field]] of ''F''(''x'') = (x<sup>3</sup>/3)-(x<sup>2</sup>/2)-x+c, showing three of the infinitely many solutions that can be produced by varying the [[Constant of integration|arbitrary constant]] ''C''.]]
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| {{Calculus |Integral}}
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| In [[calculus]], an '''antiderivative''', '''primitive [[integral]]''' or '''indefinite [[integral]]'''<ref>
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| Antiderivatives are also called '''general integrals''', and sometimes '''integrals'''. The latter term is generic, and refers not only to indefinite integrals (antiderivatives), but also to [[integral|definite integrals]]. When the word integral is used without additional specification, the reader is supposed to deduce from the context whether it is referred to a definite or indefinite integral. Some authors define the indefinite integral of a function as the set of its infinitely many possible antiderivatives. Others define it as an arbitrarily selected element of that set. Wikipedia adopts the latter approach.
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| </ref>
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| of a [[function (mathematics)|function]] ''f'' is a differentiable function ''F'' whose [[derivative]] is equal to ''f'', i.e., ''F'' ′ = ''f''.<ref>{{cite book | last=Stewart | first=James | authorlink=James Stewart (mathematician) | title=Calculus: Early Transcendentals |publisher=[[Brooks/Cole]] | edition=6th | year=2008 | isbn=0-495-01166-5}}</ref><ref>{{cite book | last1=Larson | first1=Ron | authorlink=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=[[Brooks/Cole]] | edition=9th | year=2009 | isbn=0-547-16702-4}}</ref> 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 related to [[integral|definite integral]]s through the [[fundamental theorem of calculus]]: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
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| The discrete equivalent of the notion of antiderivative is [[antidifference]]. | |
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| ==Example==
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| The function ''F''(''x'') = ''x''<sup>3</sup>/3 is an antiderivative of ''f''(''x'') = ''x''<sup>2</sup>. As the derivative of a [[Constant function|constant]] is [[0 (number)|zero]], ''x''<sup>2</sup> will have an [[infinity|infinite]] number of antiderivatives; such as (''x''<sup>3</sup>/3) + 0, (''x''<sup>3</sup>/3) + 7, (''x''<sup>3</sup>/3) − 42, (''x''<sup>3</sup>/3) + 293 etc. Thus, all the antiderivatives of ''x''<sup>2</sup> can be obtained by changing the value of C in ''F''(''x'') = (''x''<sup>3</sup>/3) + ''C''; where ''C'' is an arbitrary constant known as the [[constant of integration]]. Essentially, the [[graph of a function|graphs]] of antiderivatives of a given function are [[vertical translation]]s of each other; each graph's vertical location depending upon the [[Value (mathematics)|value]] of ''C''.
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| In physics, the integration of acceleration yields velocity 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).
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| ==Uses and properties==
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| Antiderivatives are important because they can be used to [[integral#Computing integrals|compute definite integrals]], using the [[fundamental theorem of calculus]]: if ''F'' is an antiderivative of the [[Riemann integral|integrable]] function ''f'' and ''f'' is [[continuous function|continuous]] over the interval {{nowrap|[''a'', ''b''],}} then:
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| :<math>\int_a^b f(x)\,\mathrm{d}x = F(b) - F(a).</math>
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| Because of this, each of the infinitely many 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:
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| :<math>\int f(x)\, \mathrm{d}x.</math>
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| If ''F'' is an antiderivative of ''f'', and the function ''f'' is defined on some [[interval (mathematics)|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
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| :<math>F(x)=\begin{cases}-\frac{1}{x}+C_1\quad x<0\\-\frac{1}{x}+C_2\quad x>0\end{cases}</math>
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| is the most general antiderivative of <math>f(x)=1/x^2</math> on its natural domain <math>(-\infty,0)\cup(0,\infty).</math>
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| Every [[continuous function]] ''f'' has an antiderivative, and one antiderivative ''F'' is given by the definite integral of ''f'' with variable upper boundary:
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| :<math>F(x)=\int_0^x f(t)\,\mathrm{d}t.</math>
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| Varying the lower boundary produces other antiderivatives (but not necessarily all possible antiderivatives). This is another formulation of the [[fundamental theorem of calculus]].
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| There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of [[elementary function]]s (like [[polynomial]]s, [[exponential function]]s, [[logarithm]]s, [[trigonometric functions]], [[inverse trigonometric functions]] and their combinations). Examples of these are
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| :<math>\int e^{-x^2}\,\mathrm{d}x,\qquad \int \sin x^2\,\mathrm{d}x, \qquad\int \frac{\sin x}{x}\,\mathrm{d}x,\qquad \int\frac{1}{\ln x}\,\mathrm{d}x,\qquad \int x^{x}\,\mathrm{d}x.</math>
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| See also [[differential Galois theory]] for a more detailed discussion.
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| == Techniques of integration ==
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| 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 function (differential algebra)|elementary functions]] for further information.
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| There are various methods available:
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| * the [[linearity of integration]] allows us to break complicated integrals into simpler ones
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| * [[integration by substitution]], often combined with [[trigonometric identity|trigonometric identities]] or the [[natural logarithm]]
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| * [[integration by parts]] to integrate products of functions
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| * the [[inverse chain rule method]], a special case of integration by substitution
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| * [[Inverse function integration]], a formula that expresses the antiderivative of the inverse <math>f^{-1}</math> of an invertible and continuous function <math>f</math> in terms of the antiderivative of <math>f</math> and of <math>f^{-1}</math>.
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| * the method of [[partial fractions in integration]] allows us to integrate all [[rational function]]s (fractions of two polynomials)
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| * the [[Risch algorithm]]
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| * integrals can also be looked up in a [[table of integrals]]
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| * when integrating multiple times, certain additional techniques can be used, see for instance [[double integral]]s and [[Polar coordinate system|polar coordinates]], the [[Jacobian matrix and determinant|Jacobian]] and the [[Stokes' theorem]]
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| * [[computer algebra system]]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
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| * if a function has no elementary antiderivative (for instance, exp(-''x''<sup>2</sup>)), its definite integral can be approximated using [[numerical integration]]
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| * it is often convenient to algebraically manipulate the integrand such that other integration techniques, such as integration by substitution, may be used.
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| * to calculate the (''n'' times) repeated antiderivative of a function ''f'', [[Cauchy]]'s formula is useful (cf. [[Cauchy formula for repeated integration]]):
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| ::<math>\int_{x_0}^x \int_{x_0}^{x_1} \dots \int_{x_0}^{x_{n-1}} f(x_n) \,\mathrm{d}x_n \dots \, \mathrm{d} x_2\, \mathrm{d} x_1= \int_{x_0}^x f(t) \frac{(x-t)^{n-1}}{(n-1)!}\,\mathrm{d}t .</math>
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| == Antiderivatives of non-continuous functions ==
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| Non-continuous functions can have antiderivatives. While there are still open questions in this area, it is known that:
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| * Some highly [[pathological (mathematics)|pathological functions]] with large sets of discontinuities may nevertheless have antiderivatives.
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| * In some cases, the antiderivatives of such pathological functions may be found by [[Riemann integral|Riemann integration]], while in other cases these functions are not Riemann integrable.
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| Assuming that the domains of the functions are open intervals:
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| * A necessary, but not sufficient, condition for a function ''f'' to have an antiderivative is that ''f'' have the [[intermediate value theorem|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
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| :<math>g (x) = F(x) - dx</math>
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| on the closed interval [''a'', ''b'']. Then ''g'' must have either a maximum or minimum ''c'' in the open interval (''a'', ''b'') and so
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| :<math>0 = g'(c) = f (c) - d. </math>
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| * The set of discontinuities of ''f'' must be a [[Baire space#Historical definition|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.
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| * If ''f'' has an antiderivative, is [[bounded function|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 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.
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| * If ''f'' has an antiderivative ''F'' on a closed interval [''a'',''b''], then for any choice of partition <math>a=x_0<x_1<x_2<\dots<x_n=b</math>, if one chooses sample points <math>x_i^*\in[x_{i-1},x_i]</math> as specified by the [[mean value theorem]], then the corresponding Riemann sum [[telescoping series|telescopes]] to the value ''F''(''b'') − ''F''(''a'').
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| :: <math>
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| \begin{align}
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| \sum_{i=1}^n f(x_i^*)(x_i-x_{i-1}) & = \sum_{i=1}^n [F(x_i)-F(x_{i-1})] \\
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| & = F(x_n)-F(x_0) = F(b)-F(a)
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| \end{align}
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| </math>
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| :However if ''f'' is unbounded, or if ''f'' is bounded but the set of discontinuities of ''f'' has positive Lebesgue measure, a different choice of sample points <math>x_i^*</math> may give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below.
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| === Some examples ===
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| {{ordered list
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| |1= The function
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| :<math>f(x)=2x\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x}\right)</math>
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| with <math>f\left(0\right)=0</math> is not continuous at <math>x=0</math> but has the antiderivative
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| :<math>F\left(x\right)=x^2\sin\left(\frac{1}{x}\right)</math> | |
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| with <math>F\left(0\right)=0</math>. Since ''f'' is bounded on closed finite intervals and is only discontinuous at 0, the antiderivative ''F'' may be obtained by integration: <math>F(x)=\int_0^x f(t)\,\mathrm{d}t</math>.
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| |2= The function
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| :<math>f(x)=2x\sin\left(\frac{1}{x^2}\right)-\frac{2}{x}\cos\left(\frac{1}{x^2}\right)</math>
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| with <math>f\left(0\right)=0</math> is not continuous at <math>x=0</math> but has the antiderivative
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| :<math>F(x)=x^2\sin\left(\frac{1}{x^2}\right)</math>
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| with <math>F\left(0\right)=0</math>. Unlike Example 1, ''f''(''x'') is unbounded in any interval containing 0, so the Riemann integral is undefined.
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| |3= If ''f''(''x'') is the function in Example 1 and ''F'' is its antiderivative, and <math>\{x_n\}_{n\ge1}</math> is a [[dense set|dense]] [[countable]] subset of the open interval <math>\left(-1,1\right)</math>, then the function
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| :<math>g(x)=\sum_{n=1}^\infty \frac{f(x-x_n)}{2^n}</math>
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| has an antiderivative
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| :<math>G(x)=\sum_{n=1}^\infty \frac{F(x-x_n)}{2^n}.</math>
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| The set of discontinuities of ''g'' is precisely the set <math>\{x_n\}_{n\ge1}</math>. Since ''g'' is bounded on closed finite intervals and the set of discontinuities has measure 0, the antiderivative ''G'' may be found by integration.
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| |4= Let <math>\{x_n\}_{n\ge1}</math> be a [[dense set|dense]] [[countable]] subset of the open interval <math>\left(-1,1\right)</math>. Consider the everywhere continuous strictly increasing function
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| :<math>F(x)=\sum_{n=1}^\infty\frac{1}{2^n}(x-x_n)^{1/3}.</math>
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| It can be shown that
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| :<math>F'(x)=\sum_{n=1}^\infty\frac{1}{3\cdot2^n}(x-x_n)^{-2/3}</math>
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| [[Image:antideriv1.gif|125px|right|thumb|Figure 1.]]
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| [[Image:antideriv2.gif|thumb|right|125px|Figure 2.]]
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| 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 <math>\{x_n\}_{n\ge1}</math>. | |
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| Moreover <math>F\left(x\right)\ge0</math> for all ''x'' where the derivative is defined. It follows that the inverse function <math>G=F^{-1}</math> is differentiable everywhere and that
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| :<math>g\left(x\right)=G'\left(x\right)=0</math>
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| for all ''x'' in the set <math>\{F(x_n)\}_{n\ge1}</math> which is dense in the interval <math>\left[F\left(-1\right),F\left(1\right)\right]</math>. Thus ''g'' has an antiderivative ''G''. On the other hand, it can not be true that
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| :<math>\int_{F(-1)}^{F(1)}g(x)\,\mathrm{d}x=GF(1)-GF(-1)=2,</math>
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| since for any partition of <math>\left[F\left(-1\right),F\left(1\right)\right]</math>, one can choose sample points for the Riemann sum from the set <math>\{F(x_n)\}_{n\ge1}</math>, 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 <math>\{x_n=\cos(n)\}_{n\ge1}</math> 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.
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| |5= In Examples 3 and 4, the sets of discontinuities of the functions ''g'' are dense only in a finite open interval <math>\left(a,b\right)</math>. However these examples can be easily modified so as to have sets of discontinuities which are dense on the entire real line <math>(-\infty,\infty)</math>. Let
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| :<math>\lambda(x) = \frac{a+b}{2} + \frac{b-a}{\pi}\tan^{-1} x.</math>
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| Then <math>g\left(\lambda(x)\right)\lambda'(x)</math> has a dense set of discontinuities on <math>(-\infty,\infty)</math> and has antiderivative <math>G\cdot\lambda.</math>
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| |6= 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 <math>\left[a,b\right]</math> is 0 whenever ''a'' and ''b'' are both rational, instead of <math>G\left(b\right)-G\left(a\right)</math>. Thus the fundamental theorem of calculus will fail spectacularly.
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| }}
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| ==See also==
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| * [[Antiderivative (complex analysis)]]
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| * [[Lists of integrals]]
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| * [[Symbolic integration]]
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| == Notes ==
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| <div class="references">
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| <references />
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| </div>
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| == References ==
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| * ''Introduction to Classical Real Analysis'', by Karl R. Stromberg; Wadsworth, 1981 (see [http://groups.google.com/group/sci.math/browse_frm/thread/8d900a2d79429d43/0ba4ff0d46efe076?lnk=st&q=&rnum=19&hl=en#0ba4ff0d46efe076 also])
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| * ''Historical Essay On Continuity Of Derivatives'', by Dave L. Renfro; http://groups.google.com/group/sci.math/msg/814be41b1ea8c024
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| ==External links==
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| * [http://integrals.wolfram.com Wolfram Integrator] — Free online symbolic integration with [[Mathematica]]
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| * [http://calculus-calculator.com/integral/ Online antiderivative calculator which shows the intermediate steps of calculation]
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| * [http://user.mendelu.cz/marik/maw/index.php?lang=en&form=integral Mathematical Assistant on Web] — symbolic computations online. Allows to integrate in small steps (with hints for next step (integration by parts, substitution, partial fractions, application of formulas and others), powered by [[Maxima (software)|Maxima]]
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| * [http://wims.unice.fr/wims/wims.cgi?module=tool/analysis/function.en Function Calculator] from [http://wims.unice.fr WIMS]
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| * [http://hyperphysics.phy-astr.gsu.edu/hbase/integ.html Integral]
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| * [http://www.khanacademy.org/video/the-indefinite-integral-or-anti-derivative "The Indefinite Integral or Anti-derivative "] at the [[Khan Academy]]
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| * [http://www.symbolab.com/solver/integral-calculator "Free online integral calculator with step by step solution "]
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| [[Category:Integral calculus]]
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| [[Category:Linear operators in calculus]]
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| {{Link FA|km}}
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