Английская Википедия:Antiderivative

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Файл:Slope Field.png

The slope field of <math>F(x) = \frac{x^3}{3} - \frac{x^2}{2} - x + c</math>, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant Шаблон:Mvar.

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function Шаблон:Math is a differentiable function Шаблон:Math whose derivative is equal to the original function Шаблон:Math. This can be stated symbolically as Шаблон:Math. 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 Шаблон:Mvar and Шаблон:Mvar.

Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed 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.

In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, velocity and acceleration). The discrete equivalent of the notion of antiderivative is antidifference.

Examples

The function <math>F(x) = \tfrac{x^3}{3}</math> is an antiderivative of <math>f(x) = x^2</math>, since the derivative of <math>\tfrac{x^3}{3}</math> is <math>x^2</math>. And since the derivative of a constant is zero, <math>x^2</math> will have an infinite number of antiderivatives, such as <math>\tfrac{x^3}{3}, \tfrac{x^3}{3}+1, \tfrac{x^3}{3}-2</math>, etc. Thus, all the antiderivatives of <math>x^2</math> can be obtained by changing the value of Шаблон:Math in <math>F(x) = \tfrac{x^3}{3}+c</math>, where Шаблон:Math 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, with each graph's vertical location depending upon the value Шаблон:Math.

More generally, the power function <math>f(x) = x^n</math> has antiderivative <math>F(x) = \tfrac{x^{n+1}}{n+1} + c</math> if Шаблон:Math, and <math>F(x) = \log |x| + c</math> if Шаблон:Math.

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). Thus, integration produces the relations of acceleration, velocity and displacement: <math display="block">\begin{align} \int a \, \mathrm{d}t &= v + C \\ \int v \, \mathrm{d}t &= s + C \end{align}</math>

Uses and properties

Antiderivatives can be used to compute definite integrals, using the fundamental theorem of calculus: if Шаблон:Math is an antiderivative of the continuous function Шаблон:Math over the interval <math>[a,b]</math>, then: <math display="block">\int_a^b f(x)\,\mathrm{d}x = F(b) - F(a).</math>

Because of this, each of the infinitely many antiderivatives of a given function Шаблон:Math may be called the "indefinite integral" of f and written using the integral symbol with no bounds: <math display="block">\int f(x)\,\mathrm{d}x.</math>

If Шаблон:Math is an antiderivative of Шаблон:Math, and the function Шаблон:Math is defined on some interval, then every other antiderivative Шаблон:Math of Шаблон:Math differs from Шаблон:Math by a constant: there exists a number Шаблон:Math such that <math>G(x) = F(x)+c</math> for all Шаблон:Math. Шаблон:Math is called the constant of integration. If the domain of Шаблон:Math is a disjoint union of two or more (open) intervals, then a different constant of integration may be chosen for each of the intervals. For instance <math display="block">F(x) = \begin{cases} -\dfrac{1}{x} + c_1 & x<0 \\[1ex] -\dfrac{1}{x} + c_2 & x>0 \end{cases}</math>

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

Every continuous function Шаблон:Math has an antiderivative, and one antiderivative Шаблон:Math is given by the definite integral of Шаблон:Math with variable upper boundary: <math display="block">F(x) = \int_a^x f(t)\,\mathrm{d}t ~,</math> for any Шаблон:Math in the domain of Шаблон:Math. Varying the lower boundary produces other antiderivatives, but not necessarily all possible antiderivatives. This is another formulation of the fundamental theorem of calculus.

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 Шаблон:Div col

• the error function <math display="block">\int e^{-x^2}\,\mathrm{d}x,</math>
• the Fresnel function <math display="block">\int \sin x^2\,\mathrm{d}x,</math>
• the sine integral <math display="block">\int \frac{\sin x}{x}\,\mathrm{d}x,</math>
• the logarithmic integral function <math display="block">\int\frac{1}{\log x}\,\mathrm{d}x,</math> and
• sophomore's dream <math display="block">\int x^{x}\,\mathrm{d}x.</math>

Шаблон:Div col end 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, often combined with trigonometric identities or the natural logarithm
• The inverse 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 inverse Шаблон:Math of an invertible and continuous function Шаблон:Mvar, in terms of the antiderivative of Шаблон:Mvar and of Шаблон:Math).
• The method of partial fractions in integration (which allows us to integrate all rational functions—fractions of two polynomials)
• The Risch algorithm
• Additional techniques for multiple integrations (see for instance double integrals, polar coordinates, the Jacobian and the Stokes' theorem)
• Numerical integration (a technique for approximating a definite integral when no elementary antiderivative exists, as in the case of Шаблон:Math)
• 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 Шаблон:Math-times antiderivative of a function) <math display="block"> \int_{x_0}^x \int_{x_0}^{x_1} \cdots \int_{x_0}^{x_{n-1}} f(x_n) \,\mathrm{d}x_n \cdots \, \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>

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 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 Шаблон:Math to have an antiderivative is that Шаблон:Math have the intermediate value property. That is, if Шаблон:Math is a subinterval of the domain of Шаблон:Math and Шаблон:Math is any real number between Шаблон:Math and Шаблон:Math, then there exists a Шаблон:Mvar between Шаблон:Mvar and Шаблон:Mvar such that Шаблон:Math. This is a consequence of Darboux's theorem.
• The set of discontinuities of Шаблон:Math 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 Шаблон:Math having an antiderivative, which has the given set as its set of discontinuities.
• If Шаблон:Math 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 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 Шаблон:Math has an antiderivative Шаблон:Math on a closed interval <math>[a,b]</math>, 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 telescopes to the value <math>F(b)-F(a)</math>. <math display="block">\begin{align}

\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{align}</math> However if Шаблон:Math is unbounded, or if Шаблон:Math is bounded but the set of discontinuities of Шаблон:Math 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.

Basic formulae

• If <math>{\mathrm{d} \over \mathrm{d}x} f(x) = g(x)</math>, then <math>\int g(x) \mathrm{d}x = f(x) + C</math>.
• <math>\int 1\ \mathrm{d}x = x + C</math>
• <math>\int a\ \mathrm{d}x = ax + C</math>
• <math>\int x^n \mathrm{d}x = \frac{x^{n+1}}{n+1} + C;\ n \neq -1</math>
• <math>\int \sin{x}\ \mathrm{d}x = -\cos{x} + C</math>
• <math>\int \cos{x}\ \mathrm{d}x = \sin{x} + C</math>
• <math>\int \sec^2{x}\ \mathrm{d}x = \tan{x} + C</math>
• <math>\int \csc^2{x}\ \mathrm{d}x = -\cot{x} + C</math>
• <math>\int \sec{x}\tan{x}\ \mathrm{d}x = \sec{x} + C</math>
• <math>\int \csc{x}\cot{x}\ \mathrm{d}x = -\csc{x} + C</math>
• <math>\int \frac{1}{x}\ \mathrm{d}x = \log|x| + C</math>
• <math>\int \mathrm{e}^{x} \mathrm{d}x = \mathrm{e}^{x} + C</math>
• <math>\int a^{x} \mathrm{d}x = \frac{a^{x}}{\log a} + C;\ a > 0,\ a \neq 1</math>

See also

• Antiderivative (complex analysis)
• Formal antiderivative
• Jackson integral
• Lists of integrals
• Symbolic integration
• Area

Further reading

• Introduction to Classical Real Analysis, by Karl R. Stromberg; Wadsworth, 1981 (see 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
• Integral 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