Tavola degli integrali indefiniti di funzioni d'arco

Questa pagina contiene una tavola di integrali indefiniti di funzioni d'arco. Per altri integrali vedi Integrale § Tavole di integrali.

In questa pagina si assume che c denoti una costante diversa da 0.

$\int \arcsin \frac{x}{c}dx=x\arcsin \frac{x}{c}+\sqrt{c^2-x^2}$

$\int x\arcsin \frac{x}{c}dx=(\frac{x^2}{2}-\frac{c^2}{4})\arcsin \frac{x}{c}+\frac{x}{4}\sqrt{c^2-x^2}$

$\int x^2\arcsin \frac{x}{c}dx=\frac{x^3}{3}\arcsin \frac{x}{c}+\frac{x^2+2c^2}{9}\sqrt{c^2-x^2}$

$\int \arccos \frac{x}{c}dx=x\arccos \frac{x}{c}-\sqrt{c^2-x^2}$

$\int x\arccos \frac{x}{c}dx=(\frac{x^2}{2}-\frac{c^2}{4})\arccos \frac{x}{c}-\frac{x}{4}\sqrt{c^2-x^2}$

$\int x^2\arccos \frac{x}{c}dx=\frac{x^3}{3}\arccos \frac{x}{c}-\frac{x^2+2c^2}{9}\sqrt{c^2-x^2}$

$\int \arctan \frac{x}{c}dx=x\arctan \frac{x}{c}-\frac{c}{2}\ln (c^2+x^2)$

$\int x\arctan \frac{x}{c}dx=\frac{c^2+x^2}{2}\arctan \frac{x}{c}-\frac{cx}{2}$

$\int x^2\arctan \frac{x}{c}dx=\frac{x^3}{3}\arctan \frac{x}{c}-\frac{c{x}^2}{6}+\frac{c^3}{6}\ln c^2+x^2$

$\int x^n\arctan \frac{x}{c}dx=\frac{x^{n+1}}{n+1}\arctan \frac{x}{c}-\frac{c}{n+1}\int \frac{x^{n+1}dx}{c^2+x^2}\text{(per }n\ne 1\text{)}$

$\int \mathrm{arcsec}\frac{x}{c}dx=x\mathrm{arcsec}\frac{x}{c}+\frac{x}{c|x|}\ln |x\pm \sqrt{x^2-1}|$

$\int \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{c}{2}\ln (c^2+x^2)$

$\int x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=\frac{c^2+x^2}{2}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{cx}{2}$

$\int x^2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=\frac{x^3}{3}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{c{x}^2}{6}-\frac{c^3}{6}\ln (c^2+x^2)$

$\int x^n\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=\frac{x^{n+1}}{n+1}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{c}{n+1}\int \frac{x^{n+1}dx}{c^2+x^2}\text{(per }n\ne 1\text{)}$

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