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Antiderivative

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Indefinite integral

This article is about antiderivatives in real analysis. For complex functions, seeAntiderivative (complex analysis). For lists of antiderivatives of common functions, seeLists of integrals.

Theslope field of $F(x)={\frac {x^{3}}{3}}-{\frac {x^{2}}{2}}-x+C$ , showing three of the infinitely many solutions that can be produced by varying thearbitrary constantC.

{\displaystyle F(x)={\frac {x^{3}}{3}}-{\frac {x^{2}}{2}}-x+C} — Theslope field of $F(x)={\frac {x^{3}}{3}}-{\frac {x^{2}}{2}}-x+C$ , showing three of the infinitely many solutions that can be produced by varying thearbitrary constantC.

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\int _{a}^{b}f'(t)\,dt=f(b)-f(a)

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Incalculus, anantiderivative,inverse derivative,primitive function,primitive integral orindefinite integral^{[Note 1]} of afunctionf is adifferentiable functionF whosederivative is equal to the original functionf. This can be stated symbolically asF' =f.^[1]^[2] The process of solving for antiderivatives is calledantidifferentiation (orindefinite integration), and its opposite operation is calleddifferentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capitalRoman letters such asF andG.

Antiderivatives are related todefinite integrals through thesecond fundamental theorem of calculus: the definite integral of a function over aclosed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

Inphysics, antiderivatives arise in the context ofrectilinear motion (e.g., in explaining the relationship betweenposition,velocity andacceleration).^[3] Thediscrete equivalent of the notion of antiderivative isantidifference.

Examples

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The function $F(x)={\tfrac {x^{3}}{3}}$ is an antiderivative of $f(x)=x^{2}$ , since the derivative of ${\tfrac {x^{3}}{3}}$ is $x^{2}$ . Since the derivative of aconstant iszero, $x^{2}$ will have aninfinite number of antiderivatives, such as ${\tfrac {x^{3}}{3}},{\tfrac {x^{3}}{3}}+1,{\tfrac {x^{3}}{3}}-2$ , etc. Thus, all the antiderivatives of $x^{2}$ can be obtained by changing the value ofC in $F(x)={\tfrac {x^{3}}{3}}+C$ , whereC is an arbitrary constant known as theconstant of integration. Thegraphs of antiderivatives of a given function arevertical translations of each other, with each graph's vertical location depending upon thevalueC.

More generally, thepower function $f(x)=x^{n}$ has antiderivative $F(x)={\tfrac {x^{n+1}}{n+1}}+C$ ifn ≠ −1, and $F(x)=\ln |x|+C$ ifn = −1.

Inphysics, the integration ofacceleration yieldsvelocity 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).^[3] Thus, integration produces the relations of acceleration, velocity anddisplacement: ${\begin{aligned}\int a\,dt&=v+v_{0}\\\int v\,dt&=s+s_{0}\end{aligned}}$

Uses and properties

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Antiderivatives can be used tocompute definite integrals, using thefundamental theorem of calculus: ifF is an antiderivative of thecontinuous functionf over the interval $[a,b]$ , then: $\int _{a}^{b}f(x)\,dx=F(b)-F(a).$

Because of this, each of the infinitely many antiderivatives of a given functionf may be called the "indefinite integral" off and written using the integral symbol with no bounds: $\int f(x)\,dx.$

IfF is an antiderivative off, and the functionf is defined on some interval, then every other antiderivativeG off differs fromF by a constant: there exists a numberc such that $G(x)=F(x)+c$ for allx.c is called theconstant of integration. If the domain ofF is adisjoint union of two or more (open) intervals, then a different constant of integration may be chosen for each of the intervals. For instance $F(x)={\begin{cases}-{\dfrac {1}{x}}+c_{1}&x<0\\[1ex]-{\dfrac {1}{x}}+c_{2}&x>0\end{cases}}$

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

Everycontinuous functionf has an antiderivative, and one antiderivativeF is given by the definite integral off with variable upper boundary: $F(x)=\int _{a}^{x}f(t)\,dt~,$ for anya in the domain off. Varying the lower boundary produces other antiderivatives, but not necessarily all possible antiderivatives. This is another formulation of thefundamental theorem of calculus.

There are manyelementary functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions. Elementary functions arepolynomials,exponential functions,logarithms,trigonometric functions,inverse trigonometric functions and their combinations under composition andlinear combination. Examples of thesenonelementary integrals are

theerror function $\int e^{-x^{2}}\,dx,$
theFresnel function $\int \sin x^{2}\,dx,$
thesine integral $\int {\frac {\sin x}{x}}\,dx,$
thelogarithmic integral function $\int {\frac {1}{\log x}}\,dx,$ and
sophomore's dream $\int x^{x}\,dx.$

For a more detailed discussion, see alsoDifferential Galois theory.

Techniques of integration

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Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives (indeed, there is no pre-defined method for computing indefinite integrals).^[4] For some elementary functions, it is impossible to find an antiderivative in terms of other elementary functions. To learn more, seeelementary functions andnonelementary integral.

There exist many properties and techniques for finding antiderivatives. These include, among others:

Thelinearity of integration (which breaks complicated integrals into simpler ones)
Integration by substitution, often combined withtrigonometric identities or thenatural logarithm
Theinverse chain rule method (a special case of integration by substitution)
Integration by parts (to integrate products of functions)
Inverse function integration (a formula that expresses the antiderivative of the inversef⁻¹ of an invertible and continuous functionf, in terms off⁻¹ and the antiderivative off).
The method ofpartial fractions in integration (which allows us to integrate allrational functions—fractions of two polynomials)
TheRisch algorithm
Additional techniques for multiple integrations (see for instancedouble integrals,polar coordinates, theJacobian and theStokes' theorem)
Numerical integration (a technique for approximating a definite integral when no elementary antiderivative exists, as in the case ofexp(−x²))
Algebraic manipulation of integrand (so that other integration techniques, such as integration by substitution, may be used)
Cauchy formula for repeated integration (to calculate then-times antiderivative of a function) $\int _{x_{0}}^{x}\int _{x_{0}}^{x_{1}}\cdots \int _{x_{0}}^{x_{n-1}}f(x_{n})\,dx_{n}\cdots \,dx_{2}\,dx_{1}=\int _{x_{0}}^{x}f(t){\frac {(x-t)^{n-1}}{(n-1)!}}\,dt.$

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. Integrals which have already been derived can be looked up in atable of integrals.

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:

Some highlypathological functions with large sets of discontinuities may nevertheless have antiderivatives.
In some cases, the antiderivatives of such pathological functions may be found byRiemann 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 functionf to have an antiderivative is thatf have theintermediate value property. That is, if[a,b] is a subinterval of the domain off andy is any real number betweenf(a) andf(b), then there exists ac betweena andb such thatf(c) =y. This is a consequence ofDarboux's theorem.
The set of discontinuities off must be ameagre set. This set must also be anF-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 functionf having an antiderivative, which has the given set as its set of discontinuities.
Iff has an antiderivative, isbounded on closed finite subintervals of the domain and has a set of discontinuities ofLebesgue measure 0, then an antiderivative may be found by integration in the sense of Lebesgue. In fact, using more powerful integrals like theHenstock–Kurzweil integral, every function for which an antiderivative exists is integrable, and its general integral coincides with its antiderivative.
Iff has an antiderivativeF 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 themean value theorem, then the corresponding Riemann sumtelescopes to the value $F(b)-F(a)$ . ${\begin{aligned}\sum _{i=1}^{n}f(x_{i}^{*})(x_{i}-x_{i-1})&=\sum _{i=1}^{n}[F(x_{i})-F(x_{i-1})]\\&=F(x_{n})-F(x_{0})=F(b)-F(a)\end{aligned}}$ However, iff is unbounded, or iff is bounded but the set of discontinuities off 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

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The function
$f(x)=2x\sin \left({\frac {1}{x}}\right)-\cos \left({\frac {1}{x}}\right)$ with $f(0)=0$ is not continuous at $x=0$ but has the antiderivative $F(x)=x^{2}\sin \left({\frac {1}{x}}\right)$
with $F(0)=0$ . Sincef is bounded on closed finite intervals and is only discontinuous at 0, the antiderivativeF may be obtained by integration: $F(x)=\int _{0}^{x}f(t)\,dt$ .
The function $f(x)=2x\sin \left({\frac {1}{x^{2}}}\right)-{\frac {2}{x}}\cos \left({\frac {1}{x^{2}}}\right)$ with $f(0)=0$ is not continuous at $x=0$ but has the antiderivative $F(x)=x^{2}\sin \left({\frac {1}{x^{2}}}\right)$ with $F(0)=0$ . Unlike Example 1,f(x) is unbounded in any interval containing 0, so the Riemann integral is undefined.
Iff(x) is the function in Example 1 andF is its antiderivative, and $\{x_{n}\}_{n\geq 1}$ is adense countable subset of the open interval $(-1,1),$ then the function $g(x)=\sum _{n=1}^{\infty }{\frac {f(x-x_{n})}{2^{n}}}$ has an antiderivative $G(x)=\sum _{n=1}^{\infty }{\frac {F(x-x_{n})}{2^{n}}}.$ The set of discontinuities ofg is precisely the set $\{x_{n}\}_{n\geq 1}$ . Sinceg is bounded on closed finite intervals and the set of discontinuities has measure 0, the antiderivativeG may be found by integration.
Let $\{x_{n}\}_{n\geq 1}$ be adense countable subset of the open interval $(-1,1).$ Consider the everywhere continuous strictly increasing function $F(x)=\sum _{n=1}^{\infty }{\frac {1}{2^{n}}}(x-x_{n})^{1/3}.$ It can be shown that $F'(x)=\sum _{n=1}^{\infty }{\frac {1}{3\cdot 2^{n}}}(x-x_{n})^{-2/3}$
Figure 1.
Figure 2.
for all valuesx where the series converges, and that the graph ofF(x) has vertical tangent lines at all other values ofx. In particular the graph has vertical tangent lines at all points in the set $\{x_{n}\}_{n\geq 1}$ .
Moreover $F(x)\geq 0$ for allx where the derivative is defined. It follows that the inverse function $G=F^{-1}$ is differentiable everywhere and that $g(x)=G'(x)=0$
for allx in the set $\{F(x_{n})\}_{n\geq 1}$ which is dense in the interval $[F(-1),F(1)].$ Thusg has an antiderivativeG. 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 $[F(-1),F(1)]$ , one can choose sample points for the Riemann sum from the set $\{F(x_{n})\}_{n\geq 1}$ , giving a value of 0 for the sum. It follows thatg has a set of discontinuities of positive Lebesgue measure. Figure 1 on the right shows an approximation to the graph ofg(x) where $\{x_{n}=\cos(n)\}_{n\geq 1}$ and the series is truncated to 8 terms. Figure 2 shows the graph of an approximation to the antiderivativeG(x), also truncated to 8 terms. On the other hand if the Riemann integral is replaced by theLebesgue integral, thenFatou's lemma or thedominated convergence theorem shows thatg does satisfy the fundamental theorem of calculus in that context.
In Examples 3 and 4, the sets of discontinuities of the functionsg are dense only in a finite open interval $(a,b).$ 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 $\lambda (x)={\frac {a+b}{2}}+{\frac {b-a}{\pi }}\tan ^{-1}x.$ Then $g(\lambda (x))\lambda '(x)$ has a dense set of discontinuities on $(-\infty ,\infty )$ and has antiderivative $G\cdot \lambda .$
Using a similar method as in Example 5, one can modifyg in Example 4 so as to vanish at allrational numbers. If one uses a naive version of theRiemann 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 functiong over an interval $[a,b]$ is 0 whenevera andb are both rational, instead of $G(b)-G(a)$ . Thus the fundamental theorem of calculus will fail spectacularly.
A function which has an antiderivative may still fail to be Riemann integrable. The derivative ofVolterra's function is an example.

Basic formulae

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If ${\frac {d}{dx}}f(x)=g(x)$ , then $\int g(x)dx=f(x)+C$ .
$\int 1dx=x+C$
$\int a\ dx=ax+C$
$\int x^{n}\ dx={\frac {x^{n+1}}{n+1}}+C;\ n\neq -1$
$\int \sin {x}\ dx=-\cos {x}+C$
$\int \cos {x}\ dx=\sin {x}+C$
$\int \sec ^{2}{x}\ dx=\tan {x}+C$
$\int \csc ^{2}{x}\ dx=-\cot {x}+C$
$\int \sec {x}\tan {x}\ dx=\sec {x}+C$
$\int \csc {x}\cot {x}\ dx=-\csc {x}+C$
$\int {\frac {dx}{x}}=\ln |x|+C$
$\int e^{x}\ dx=e^{x}+C$
$\int a^{x}\ dx={\frac {a^{x}}{\ln a}}+C;\ a>0,\ a\neq 1$
$\int {\frac {1}{\sqrt {a^{2}-x^{2}}}}\ dx=\arcsin \left({\frac {x}{a}}\right)+C$
$\int {\frac {1}{a^{2}+x^{2}}}\ dx={\frac {1}{a}}\arctan \left({\frac {x}{a}}\right)+C$

Notes

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^Antiderivatives are also calledgeneral integrals, and sometimesintegrals. The latter term is generic, and refers not only to indefinite integrals (antiderivatives), but also todefinite integrals. When the wordintegral is used without additional specification, the reader is supposed to deduce from the context whether it refers 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. This article adopts the latter approach. In English A-Level Mathematics textbooks one can find the termcomplete primitive - L. Bostock and S. Chandler (1978)Pure Mathematics 1;The solution of a differential equation including the arbitrary constant is called the general solution (or sometimes the complete primitive).

References

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^Stewart, James (2008).Calculus: Early Transcendentals (6th ed.).Brooks/Cole.ISBN 978-0-495-01166-8.
^Larson, Ron; Edwards, Bruce H. (2009).Calculus (9th ed.).Brooks/Cole.ISBN 978-0-547-16702-2.
^^a ^b"4.9: Antiderivatives".Mathematics LibreTexts. 2017-04-27. Retrieved2020-08-18.
^"Antiderivative and Indefinite Integration | Brilliant Math & Science Wiki".brilliant.org. Retrieved2020-08-18.

External links

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Wolfram Integrator — Free online symbolic integration withMathematica
Function Calculator from WIMS
Integral atHyperPhysics
Antiderivatives and indefinite integrals at theKhan Academy
Integral calculator atSymbolab
The Antiderivative atMIT
Introduction to Integrals atSparkNotes
Antiderivatives at Harvy Mudd College

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