Primitives de fonctions irrationnelles

{{#invoke:Bandeau|ébauche}} Cet article dresse une liste non exhaustive de primitives de fonctions irrationnelles.

On suppose <math>a\ne 0</math>.

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

<math>\int \frac{1}{\sqrt{ax^2+bx+c}}\, \mathrm dx</math><math>=\begin{cases}

\frac{1}{\sqrt{a}}\operatorname{arsinh}\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}</math>

<math>\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</math>

<math>\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</math>

On suppose <math>a>0</math>

<math>\int \frac{1}{\sqrt{a^2-x^2}}\,\mathrm dx=\operatorname{arcsin}\frac{x}{a}+C</math>

<math>\int \frac{1}{\sqrt{a^2+x^2}}\,\mathrm dx=\operatorname{arsinh}\frac{x}{a}+C</math>

<math>\int \frac{1}{\sqrt{x^2-a^2}}\,\mathrm dx=\operatorname{arcosh}\frac{x}{a}+C</math>

<math>\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</math>

<math>\int \sqrt{a^2+x^2}\,\mathrm dx=\frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}\operatorname{arsinh}\frac{x}{a}+C</math>

<math>\int \sqrt{x^2-a^2}\,\mathrm dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\operatorname{arcosh}\frac{x}{a}+C</math>

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

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

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

<math>\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</math>

<math>\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</math>

<math>\int \frac{1}{x}\sqrt{x^2-a^2}\,\mathrm dx=\sqrt{x^2-a^2}-a\operatorname{arccos}\frac{a}{x}+C</math>

<math>\int \frac{x}{\sqrt{a^2-x^2}}\,\mathrm dx=-\sqrt{a^2-x^2}+C</math>

<math>\int \frac{x}{\sqrt{a^2+x^2}}\,\mathrm dx=\sqrt{a^2+x^2}+C</math>

<math>\int \frac{x}{\sqrt{x^2-a^2}}\,\mathrm dx=\sqrt{x^2-a^2}+C</math>

<math>\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</math>

<math>\int \frac{x^2}{\sqrt{a^2+x^2}}\,\mathrm dx=\frac{x}{2}\sqrt{a^2+x^2}-\frac{a^2}{2}\operatorname{arsinh}\frac{x}{a}+C </math>

<math>\int \frac{x^2}{\sqrt{x^2-a^2}}\,\mathrm dx=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\operatorname{arcosh}\frac{x}{a}+C </math>