Primitive

See also: Primitive (homonymy)

In Mathematical, a primitive (or, seldom, antidérivée - of English antiderivative ) of a function $f$ of a variable real is a function $F$ such as for all $x$, the Dérivée from $F\; (X)$ is equal to $f\; (X)$:

$\backslash \; forall\; X\; \backslash \; in\; \backslash \; R,\; \backslash \; quad\; F\; \backslash ,\; \text{'}(X)\; =\; F\; (X)$

A sufficient condition so that a function $f$ admits primitives on an interval is that it is there continuous.

If $f$ is a function admitting a primitive $F$ on an interval $I$, then for any reality $k$, a primitive of $kf$ on the interval $I$ is $kF$.

If $F$ and $G$ are respective primitives of two functions $f$ and $g$, then a primitive of $f+g$ is $F+G$.

If a function $f$ admits a primitive on an interval, she admits a Infini t-piece of it, which differs from a constant: if $F\_1$ and $F\_2$ are two primitives of $f$, then there exists a reality $k\_0$ such as $F\_1\; =\; F\_2+k\_0$.

If $F$ is a primitive of $f$, then

$F\; (B)\; -\; F\; (a)\; =\; \backslash \; int^b\_a\; F\; (X)\; \backslash ;\; \backslash \; mathrm\; dx$.

This is the second part of the fundamental Théorème of the analysis.

Examples

; Polynomial S and rational functions

• a primitive of the function $f\; (X)\; =\; 2x$ is $F\; (X)\; =\; x^2$
• a primitive of the function $g\; (X)\; =\; 4x^3$ is $G\; (X)\; =\; x^4$
• a primitive of the function $(f+g)\; (X)\; =\; 2x\; +\; 4x^3$ is $(F\; +\; G)\; (X)\; =\; x^2\; +\; x^4$
• a primitive of the function $f\; (X)\; =\; x^n$ is $\backslash \; tfrac\; \{x^\; \{n+1\}\}\; \{n+1\}$ for $n$ real different from −1.
• a primitive of the function reverses $f\; (X)\; =\; \backslash \; tfrac\; \{1\}\; \{X\}$ is the function Napierian logarithm $\backslash \; ln\; (X)$.
• In the general case, it does not have there in a way simple to have the primitive of a rational fraction except if one is able at the to break up into simple elements.

; goniometrical Functions

• a primitive of the cosine is the function sine.
• a primitive of the sine is the opposite of the function cosine.

; Others

• a primitive of the function Exponentielle is the exponential function itself.

Automatic calculation

Software like Maple or Mathematica has made it possible for a few years interactivement to calculate certain primitives in form symbolic system. The first software making it possible to carry out integration computer-assisted in form symbolic system was the language FORMAC, used by the physicists in the Années 1970.

Current primitives

See also: Table of primitives

For the first table, the first column is the function which one seeks the primitive, the second is its field of derivation and the third, the primitive corresponding to this function.

For the second table, the first column is the function which one seeks the primitive and the second, the primitive corresponding to the function

Simple functions

Are $a,\; B,\; C$ of the constants.

Made up functions

Are $u$ and $v$ two functions.

See too

• Integral
• Integral unsuitable
• Table of integrals
• numerical Calculation of an integral
• Table of primitives

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