Primitives de fonctions irrationnelles

Cet article dresse une liste non exhaustive des primitives de fonctions irrationnelles.

On suppose $a\ne 0$.

$\int (ax+b)^{\alpha}\,\mathrm dx=\frac{1}{(\alpha+1)a}(ax+b)^{\alpha+1}+C$ ($\alpha\ne-1$)

$\int \frac{1}{\sqrt{ax^2+bx+c}}\, \mathrm dx$$=\begin{cases} \frac{1}{\sqrt{a}}\operatorname{argsh}\frac{2ax+b}{\sqrt{-(b^2-4ac)}} +C & \text{si } b^2-4ac<0 \text{ et } a>0\\ \frac{1}{\sqrt{a}}\ln|2ax+b| +C & \text{si } b^2-4ac=0 \text{ et } a>0\\ -\frac{1}{\sqrt{-a}}\operatorname{Arcsin}\frac{2ax+b}{\sqrt{b^2-4ac}} +C &\text{si } b^2-4ac>0 \text{ et } a<0\\ \end{cases}$

$\int \sqrt{ax^2+bx+c}\,\mathrm dx=\frac{2ax+b}{4a}\sqrt{ax^2+bx+c}-\frac{b^2-4ac}{8a}\int\frac{1}{\sqrt{ax^2+bx+c}}\,\mathrm dx$

$\int \frac{x}{\sqrt{ax^2+bx+c}}\,\mathrm dx=\frac{\sqrt{ax^2+bx+c}}{a}-\frac{b}{2a}\int\frac{1}{\sqrt{ax^2+bx+c}}\,\mathrm dx$

On suppose $a>0$

$\int \frac{1}{\sqrt{a^2-x^2}}\,\mathrm dx=\operatorname{Arcsin}\frac{x}{a}+C$

$\int \frac{1}{\sqrt{a^2+x^2}}\,\mathrm dx=\operatorname{argsh}\frac{x}{a}+C$

$\int \frac{1}{\sqrt{x^2-a^2}}\,\mathrm dx=\operatorname{argch}\frac{x}{a}+C$

$\int \sqrt{a^2-x^2}\,\mathrm dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\operatorname{Arcsin}\frac{x}{a}+C$

$\int \sqrt{a^2+x^2}\,\mathrm dx=\frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}\operatorname{argsh}\frac{x}{a}+C$

$\int \sqrt{x^2-a^2}\,\mathrm dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\operatorname{argch}\frac{x}{a}+C$

$\int x\sqrt{a^2+x^2}\,\mathrm dx=\frac{1}{3}\sqrt{(a^2+x^2)^3}+C$

$\int x\sqrt{a^2-x^2}\,\mathrm dx=-\frac{1}{3}\sqrt{(a^2-x^2)^3}+C$

$\int x\sqrt{x^2-a^2}\,\mathrm dx=\frac{1}{3}\sqrt{(x^2-a^2)^3}+C$

$\int \frac{1}{x}\sqrt{a^2+x^2}\,\mathrm dx=\sqrt{a^2+x^2}-a\ln\left|\frac{1}{x}\left(a+\sqrt{a^2+x^2}\right)\right|+C$

$\int \frac{1}{x}\sqrt{a^2-x^2}\,\mathrm dx=\sqrt{a^2-x^2}-a\ln\left|\frac{1}{x}\left(a+\sqrt{a^2-x^2}\right)\right|+C$

$\int \frac{1}{x}\sqrt{x^2-a^2}\,\mathrm dx=\sqrt{x^2-a^2}-a\operatorname{Arccos}\frac{a}{x}+C$

$\int \frac{x}{\sqrt{a^2-x^2}}\,\mathrm dx=-\sqrt{a^2-x^2}+C$

$\int \frac{x}{\sqrt{a^2+x^2}}\,\mathrm dx=\sqrt{a^2+x^2}+C$

$\int \frac{x}{\sqrt{x^2-a^2}}\,\mathrm dx=\sqrt{x^2-a^2}+C$

$\int \frac{x^2}{\sqrt{a^2-x^2}}\,\mathrm dx=-\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\operatorname{Arcsin}\frac{x}{a}+C$

$\int \frac{x^2}{\sqrt{a^2+x^2}}\,\mathrm dx=\frac{x}{2}\sqrt{a^2+x^2}-\frac{a^2}{2}\operatorname{argsh}\frac{x}{a}+C$

$\int \frac{x^2}{\sqrt{x^2-a^2}}\,\mathrm dx=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\operatorname{argch}\frac{x}{a}+C$