Antiderivative

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Fundamental theorem

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File: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 Template:Mvar.

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral<ref group="Note">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 definite integrals. When the word integral 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 term complete 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). </ref> of a continuous function Template:Math is a differentiable function Template:Math whose derivative is equal to the original function Template:Math. This can be stated symbolically as Template:Math.<ref>Template:Cite book</ref><ref>Template:Cite book</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 often denoted by capital Roman letters such as Template:Mvar and Template: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).<ref name=":1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The discrete equivalent of the notion of antiderivative is antidifference.

ExamplesEdit

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>. 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 Template:Math in <math>F(x) = \tfrac{x^3}{3}+C</math>, where Template:Math 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 Template: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 Template:Math, and <math>F(x) = \ln |x| + C</math> if Template: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).<ref name=":1" /> 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 propertiesEdit

Antiderivatives can be used to compute definite integrals, using the fundamental theorem of calculus: if Template:Math is an antiderivative of the continuous function Template: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 Template: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 Template:Math is an antiderivative of Template:Math, and the function Template:Math is defined on some interval, then every other antiderivative Template:Math of Template:Math differs from Template:Math by a constant: there exists a number Template:Math such that <math>G(x) = F(x)+c</math> for all Template:Math. Template:Math is called the constant of integration. If the domain of Template: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 Template:Math has an antiderivative, and one antiderivative Template:Math is given by the definite integral of Template:Math with variable upper boundary: <math display="block">F(x) = \int_a^x f(t)\,\mathrm{d}t ~,</math> for any Template:Math in the domain of Template: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 elementary functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions. Elementary functions are polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations under composition and linear combination. Examples of these nonelementary integrals are Template: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>

Template:Div col end For a more detailed discussion, see also Differential Galois theory.

Techniques of integrationEdit

Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives (indeed, there is no pre-defined method for computing indefinite integrals).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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 Template:Math of an invertible and continuous function Template:Mvar, in terms of Template:Math and the antiderivative of Template:Mvar).
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 Template: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 Template: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 functionsEdit

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 Template:Math to have an antiderivative is that Template:Math have the intermediate value property. That is, if Template:Math is a subinterval of the domain of Template:Math and Template:Math is any real number between Template:Math and Template:Math, then there exists a Template:Mvar between Template:Mvar and Template:Mvar such that Template:Math. This is a consequence of Darboux's theorem.
The set of discontinuities of Template: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 Template:Math having an antiderivative, which has the given set as its set of discontinuities.
If Template: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 Template:Math has an antiderivative Template: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 Template:Math is unbounded, or if Template:Math is bounded but the set of discontinuities of Template: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.

Some examplesEdit

Template:Ordered list

Basic formulaeEdit

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 = \ln|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}}{\ln a} + C;\ a > 0,\ a \neq 1</math>
<math>\int \frac{1}\sqrt{a^2 - x^2}\ \mathrm{d}x = \arcsin\left(\frac{x}{a}\right) + C</math>
<math>\int \frac{1}{a^2 + x^2}\ \mathrm{d}x = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C</math>