Table de primitives

• Linéarité :
  ${\displaystyle \int \left({\color {Blue}a}\,{\color {Blue}f(x)}+{\color {blue}b}\,{\color {blue}g(x)}\right)\mathrm {d} x={\color {Blue}a}\int {\color {Blue}f(x)}\,\mathrm {d} x+{\color {Blue}b}\int {\color {Blue}g(x)}\,\mathrm {d} x}$ 
• relation de Chasles :
  ${\displaystyle \int _{a}^{c}f(x)\,\mathrm {d} x=\int _{a}^{\color {blue}b}f(x)\,\mathrm {d} x+\int _{\color {blue}b}^{c}f(x)\,\mathrm {d} x}$ 
  et en particulier :
  ${\displaystyle \int _{\color {blue}a}^{\color {blue}b}f(x)\,\mathrm {d} x={\color {blue}-}\int _{\color {blue}b}^{\color {blue}a}f(x)\,\mathrm {d} x}$ 
• intégration par parties :
  ${\displaystyle \int {\color {Blue}f(x)}\,{\color {blue}g'(x)}\,\mathrm {d} x=[{\color {Blue}f(x)}\,{\color {Blue}g(x)}]-\int {\color {Blue}f'(x)}\,{\color {blue}g(x)}\,\mathrm {d} x}$ 
  moyen mnémotechnique :
  ${\displaystyle \int {\color {Blue}u}{\color {blue}v'}=[{\color {Blue}u}{\color {Blue}v}]\ -\int {\color {Blue}u'}{\color {blue}v}}$ 

avec ${\displaystyle u=f(x),~u'=f'(x),~v=g(x),~v'=g'(x)}$  et dx implicite.

• intégration par changement de variable (si f et φ' sont continues) :
  ${\displaystyle \int _{a}^{b}f({\color {Blue}\varphi (t)})\,{\color {Blue}\varphi '(t)}\,\mathrm {d} {\color {Blue}t}=\int _{\color {blue}\varphi (a)}^{\color {blue}\varphi (b)}f({\color {Blue}x})\,\mathrm {d} {\color {Blue}x}}$ .

${\displaystyle \int \,\mathrm {d} x=C\qquad \forall x\in \mathbb {R} }$ 

${\displaystyle \int x^{n}\,\mathrm {d} x={\frac {x^{n+1}}{n+1}}+C\qquad {\text{ si }}n\neq -1}$ 

${\displaystyle \int {\frac {1}{x}}\,\mathrm {d} x=\ln \left|x\right|+C\qquad {\text{ si }}x\neq 0}$ 

${\displaystyle \int {\frac {1}{x-a}}\,\mathrm {d} x=\ln |x-a|+C\qquad {\text{ si }}x\neq a}$ 

${\displaystyle \int {\frac {1}{(x-a)^{n}}}\,\mathrm {d} x=-{\frac {1}{(n-1)(x-a)^{n-1}}}+C\qquad {\text{ si }}n\neq 1{\text{ et }}x\neq a}$ 

${\displaystyle \int {\frac {1}{1+x^{2}}}\,\mathrm {d} x=\operatorname {arctan} x+C\qquad \forall x\in \mathbb {R} }$ 

${\displaystyle \int {\frac {1}{a^{2}+x^{2}}}\,\mathrm {d} x={\frac {1}{a}}\operatorname {arctan} {\frac {x}{a}}+C\qquad {\text{ si }}a\neq 0}$ 

${\displaystyle \int {\frac {1}{1-x^{2}}}\,\mathrm {d} x={\frac {1}{2}}\ln {\left|{\frac {x+1}{x-1}}\right|}+C={\begin{cases}\operatorname {artanh} x+C&{\text{ sur }}]-1,1[\\\operatorname {arcoth} x+C&{\text{ sur }}]-\infty ,-1[{\text{ et sur }}]1,+\infty [.\end{cases}}}$ 

${\displaystyle \forall x\in \mathbb {R} _{+}^{*}}$ 

${\displaystyle \int \ln x\,\mathrm {d} x=x\ln x-x+C}$ 

Plus généralement, une primitive n-ième de ${\displaystyle \ln }$  est :

${\displaystyle {\frac {x^{n}}{n!}}\left(\ln x-\sum _{k=1}^{n}{\frac {1}{k}}\right)}$ .

${\displaystyle \forall x\in \mathbb {R} }$ 

${\displaystyle \int e^{ax}\,dx={\frac {1}{a}}e^{ax}+C}$ 

${\displaystyle \int f'(x)e^{f(x)}\,dx=e^{f(x)}+C}$ 

${\displaystyle \int a^{x}\,\mathrm {d} x={\frac {a^{x}}{\ln a}}+C\qquad {\text{ si }}a>0}$  et a ≠ 1 car ln(1) = 0.

${\displaystyle \forall x\in \mathbb {R} \setminus \{-1,1\}}$ 

${\displaystyle \int {1 \over {\sqrt {1-x^{2}}}}\,\mathrm {d} x=\operatorname {arcsin} x+C}$ 

${\displaystyle \int {-1 \over {\sqrt {1-x^{2}}}}\,\mathrm {d} x=\operatorname {arccos} x+C}$ 

${\displaystyle \int {x \over {\sqrt {x^{2}-1}}}\,\mathrm {d} x={\sqrt {x^{2}-1}}+C}$ 

• (en) Alan Jeffrey et Daniel Zwillinger, Table of Integrals, Series, and Products, Academic Press, 2007 (ISBN 978-0123736376)
• (en) Milton Abramowitz et Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [détail de l’édition] (lire en ligne)

• Calcul intégral
• Calcul numérique d'une intégrale
• Intégration
• Table d'intégrales

Calculateur automatique de primitive par Mathematica

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