Table de primitives

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

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

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

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

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

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

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

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

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

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

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