Indefinite integral

An antiderivative or an indefinite integral is a mathematical function that is investigated in differential calculus , a branch of analysis . Depending on the context, it may be necessary to distinguish between these two terms (see section "Indefinite integral").

definition

Under one common function of a real function is defined as a differentiable function whose derivative function with matches. Is therefore on a interval defined, has to be defined and differentiable, and it must for any number of valid: ${\displaystyle f}$ ${\displaystyle F,}$ ${\displaystyle F'}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle I}$ ${\displaystyle F}$ ${\displaystyle I}$ ${\displaystyle x}$ ${\displaystyle I}$

${\displaystyle F'(x)=f(x).}$

Existence and uniqueness

Every function that is continuous on an interval has an antiderivative. According to the main theorem of differential and integral calculus is namely integrable and the integral function ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle f}$

${\displaystyle x\mapsto \int _{a}^{x}f(t)\,\mathrm {d} t}$

is an antiderivative of . ${\displaystyle f}$

If on is integrable, but not continuous everywhere, then the integral function does exist, but it does not need to be differentiable at the points where is not continuous, so it is generally not an antiderivative. It is necessary for the existence of an antiderivative that the function satisfies the intermediate value theorem. This follows from the intermediate value theorem for derivatives . ${\displaystyle f}$ ${\displaystyle [a,b]}$ ${\displaystyle f}$

If a function has an antiderivative, it even has an infinite number. If namely is an antiderivative of , then for any real number the function defined by is also an antiderivative of . If the domain of is an interval, then all antiderivatives are obtained in this way: If and are two antiderivatives of , then is constant. If the domain of is not an interval, then the difference between two antiderivatives of is not necessarily constant, but is locally constant , that is, constant on every connected subset of the domain. ${\displaystyle f}$ ${\displaystyle F}$ ${\displaystyle f}$ ${\displaystyle C}$ ${\displaystyle G(x)=F(x)+C}$ ${\displaystyle G}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle f}$ ${\displaystyle G-F}$ ${\displaystyle f}$ ${\displaystyle f}$

Indefinite integral

The term of the indefinite integral is not used consistently in the specialist literature. On the one hand, the indefinite integral of is understood as a synonym for an antiderivative. The problem with this definition is that the term is nonsensical. Because in this case the indefinite integral is not a mapping because it is not clear to which of the infinitely many antiderivatives the function is to be mapped. Since the constant by which all antiderivatives differ, however, often does not play a role, this definition of the indefinite integral is not very problematic. ${\displaystyle \textstyle \int f(x)\,\mathrm {d} x}$ ${\displaystyle f}$ ${\displaystyle f\mapsto \textstyle \int f(x)\mathrm {d} x}$ ${\displaystyle f}$

Another way to understand the indefinite integral is to define the expression as the totality of all antiderivatives. This definition has the advantage that the indefinite integral is a linear mapping analogous to the definite integral, even if its values ​​are equivalence classes . ${\displaystyle \textstyle \int f(x)\,\mathrm {d} x}$

A somewhat less common way of defining the indefinite integral is to use it as a parameter integral

${\displaystyle \int _{a}^{x}f(t)\,\mathrm {d} t}$

understand. Due to the main theorem of differential and integral calculus , this expression yields an antiderivative of for every continuous function . If one extends this definition to Lebesgue integrals over arbitrary measure spaces, the indefinite integral is generally no longer an antiderivative. ${\displaystyle f}$ ${\displaystyle f}$

Examples

• An antiderivative of the polynomial function is, for example . The constant was chosen freely, in this case this antiderivative could be obtained by reversing elementary derivation rules. ${\displaystyle x^{3}+5x+6}$ ${\displaystyle {\tfrac {1}{4}}x^{4}+{\tfrac {5}{2}}x^{2}+6x+3}$ ${\displaystyle 3}$
• If one looks at the function then applies . The mapping is to an antiderivative of , but not to the whole , because it is not differentiable for.
  ${\displaystyle \operatorname {sgn} (x)={\begin{cases}-1&x<0\\1&x\geq 0,\end{cases}}}$
  ${\displaystyle \textstyle \int _{0}^{x}\operatorname {sgn} (t)\mathrm {d} t=|x|}$ ${\displaystyle x\mapsto |x|}$ ${\displaystyle \mathbb {R} \setminus \left\{0\right\}}$ ${\displaystyle \operatorname {sgn} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle |x|}$ ${\displaystyle x=0}$

application

If a function is continuous (or more generally Riemann integrable ) on the compact , i.e. finite and closed interval, the definite integral of over can be calculated with the help of any antiderivative of : ${\displaystyle f}$ ${\displaystyle [a,b]}$ ${\displaystyle F}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle [a,b]}$

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=F(b)-F(a).}$

Antiderivatives are therefore required for various calculations, e.g. B .:

• for determining the size of an area bounded by function graphs
• Volume calculation for solids of revolution

Seclusion / integration rules

There are simple rules for differentiating. In contrast, the situation with indefinite integration is completely different, since the operation of indefinite integration leads to an extension of given functional classes, e.g. B. The integration within the class of the rational functions is not completed and leads to the functions and . The class of so-called elementary functions is not closed either. Thus Joseph Liouville proved that the simple function has no elementary antiderivative. Even the simple function does not have an elementary antiderivative. Against it . ${\displaystyle \ln }$ ${\displaystyle \arctan }$ ${\displaystyle f(x)=e^{-x^{2}}}$ ${\displaystyle f(x)={\tfrac {1}{\ln x}}}$ ${\displaystyle \int {\tfrac {\ln x}{x}}\,\mathrm {d} x={\tfrac {1}{2}}\ln ^{2}x}$

The integration technique is based on the following integration rules:

• Additivity,
• the table of the so-called basic integrals (overview of the most important functions and their parent functions),
• Method of partial integration ,
• Substitution rule ,
• special procedures as well
• Use of sophisticated decompositions and transformations (e.g. polynomial division , partial fraction decomposition , functional equations ).

Since there is no general rule for determining antiderivatives, antiderivatives are tabulated in so-called integral tables. Computer algebra systems (CAS) are today able to calculate almost all integrals tabulated up to now. The Risch algorithm solves the problem of the algebraic integration of elementary functions and can decide whether an elementary antiderivative exists.

Antiderivatives for complex functions

The term antiderivative can also be formulated for complex functions. Because the derivative of a holomorphic function is holomorphic, only holomorphic functions can have antiderivatives. Holomorphy is already sufficiently locally: Is an area , a holomorphic function and then there is a neighborhood of in and a primitive function of , d. H. for everyone . ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle f\colon D\to \mathbb {C} }$ ${\displaystyle z_{0}\in D}$ ${\displaystyle U}$ ${\displaystyle z_{0}}$ ${\displaystyle D}$ ${\displaystyle F\colon U\to \mathbb {C} }$ ${\displaystyle f|U}$ ${\displaystyle F'(z)=f(z)}$ ${\displaystyle z\in U}$

The question of the existence of antiderivatives on the whole is related to topological properties of . ${\displaystyle D}$ ${\displaystyle D}$

For a holomorphic function with open and connected, the following statements are equivalent: ${\displaystyle f\colon D\to \mathbb {C} }$ ${\displaystyle D}$

1. The function has an antiderivative to the whole , that is, is holomorphic and is the complex derivative of . ${\displaystyle f}$ ${\displaystyle F}$ ${\displaystyle D}$ ${\displaystyle F}$ ${\displaystyle f}$ ${\displaystyle F}$
2. Path integrals over depend only on the end points of the way. ${\displaystyle f}$
3. Path integrals over closed paths (starting point = end point) always return 0 as the result.

For an area are equivalent: ${\displaystyle D\subseteq \mathbb {C} }$

1. Every holomorphic function has an antiderivative . ${\displaystyle f\colon D\to \mathbb {C} }$ ${\displaystyle F}$
2. Every continuous, closed path is null homotop . ${\displaystyle \gamma \colon [0,1]\to D}$
3. Every continuous, closed path is null homologous . ${\displaystyle \gamma \colon [0,1]\to D}$
4. ${\displaystyle D}$is simply connected .

See also

• Table of derivative and antiderivatives
• Convolution , for a method for interpreting and finding antiderivatives.

Web links

Wiktionary: antiderivative  - explanations of meanings, word origins, synonyms, translations

• The Integrator - Calculation of antiderivatives online
• Integral calculator with calculation path - calculation of antiderivatives with calculation path and step-by-step explanation
• Integral function applet - interactive worksheet with solutions for visualizing the concept of integral function
• Video: antiderivative, indefinite integral, main theorem . Jörn Loviscach 2011, made available by the Technical Information Library (TIB), doi : 10.5446 / 9907 .

Individual evidence

1. ↑ Harro Heuser: Textbook of Analysis. Part 1. 8th edition, BG Teubner, Stuttgart 1990. ISBN 3-519-12231-6 , chap. 76.
2. ^ Konrad Königsberger : Analysis 2. Springer-Verlag, Berlin / Heidelberg, 2000, ISBN 3-540-43580-8 , p. 201
3. ↑ Otto Forster : Analysis Volume 1: Differential and integral calculus of a variable. Vieweg-Verlag, 7th edition 2006, ISBN 3-528-67224-2 , p. 201.
4. ↑ IP Natanson: Theory of the functions of a real variable. Verlag Harry Deutscher Thun, 1981 Frankfurt am Main, ISBN 3-87144-217-8 , p. 408.
5. ^ Fritz Reinhardt, Heinrich Soeder: dtv atlas for mathematics. Volume 2, Deutscher Taschenbuch Verlag, Munich 1977, ISBN 3-423-03008-9 , p. 333.