Taula de primitives

La llista de primitives inmediates és la següent:

Taula de les integrals immediates
f(x)	$\int f (x) d x$	Exemples
k constant	$k \cdot x + C$	$\int 3 d x = 3 x + C$
$x^{n} si n \neq - 1$	$\frac{x^{n + 1}}{n + 1} + C$	$\int x^{3} d x = \frac{x^{4}}{4} + C$ $\int \sqrt{x} d x = \frac{2}{3} {(\sqrt{x})}^{3} + C$
$\frac{1}{x}$	$ln \| x \| + C$
$a^{x}$	$\frac{a^{x}}{ln (a)} + C$	$\int 3^{x} d x = \frac{3^{x}}{ln (3)} + C$ $\int e^{x} d x = e^{x} + C$
$cos (x)$	$sin (x) + C$
$sin (x)$	$- cos (x) + C$
$\frac{1}{\sqrt{1 - x^{2}}}$	$arcsin (x) + C$
$\frac{- 1}{\sqrt{1 - x^{2}}}$	$arccos (x) + C$ >
$\frac{1}{1 + x^{2}}$	$arctan (x) + C$

A partir de la regla de la cadena podem generalitzar la taula d’integrals immediates anterior a les integrals que s’anomenen sovint integrals quasi immediates

Integrals quasi immediates
Integral	Exemple
${\int [f (x)]}^{n} \cdot f' (x) d x = \frac{{[f (x)]}^{n + 1}}{n + 1} + C si n \neq 1$	$\int {[sin (x)]}^{4} \cdot cos x d x = \frac{{[sin (x)]}^{5}}{5} + C$
$\int \frac{f' (x)}{f (x)} d x = ln \| f (x) \| + C$	$\int \frac{2 x - 3}{x^{2} - 3 x + 13} d x = ln \| x^{2} - 3 x + 12 \| + C$
$\int e^{f (x)} \cdot f' (x) d x = e^{f (x)} + C$	$\int e^{4 x^{2} + 3 x - 2} \cdot (8 x + 3) d x = e^{4 x^{2} + 3 x - 2} + C$
$\int a^{f (x)} \cdot f' (x) d x = \frac{a^{f (x)}}{ln a} + C$	$\int 5^{4 x^{2} + 3 x - 2} \cdot (8 x + 3) d x = \frac{5^{4 x^{2} + 3 x - 2}}{ln 5} + C$
$\int f' (x) \cdot sin (f (x)) d x = - cos (f (x)) + C$	$\int cos x \cdot sin (sin (x)) d x = - cos (sin (x)) + C$
$\int f' (x) \cdot cos (f (x)) d x = sin (f (x)) + C$	$\int 6 \cdot cos (6 x - 2) d x = sin (6 x - 2) + C$
$\int \frac{f' (x)}{1 + {(f (x))}^{2}} d x = \arctan (f (x)) + C$	$\int \frac{\frac{1}{x}}{1 + {(ln (x))}^{2}} d x = \arctan (ln (x)) + C$
$\int \frac{f' (x)}{\sqrt{1 - {(f (x))}^{2}}} d x = arcsin (f (x)) + C$	$\int \frac{e^{x}}{\sqrt{1 - {(e^{x})}^{2}}} d x = arcsin (e^{x}) + C$