Integrálási segédlet

$$\int f(x) \dd x = F(x) + C$$

Táblázatok

Alapesetek

$f(x)$	$F(x)$
$k$	$kx + C$
$x^n$	$\dfrac{x^{n+1}}{n+1} + C$
$\dfrac{1}{x}$	$\ln \abs{x} + C$
$e^x$	$e^x + C$
$a^x$	$\dfrac{a^x}{\ln a} + C$

Szögfüggvények

$f(x)$	$F(x)$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\dfrac{1}{\cos^2 x}$	$\tan x + C$
$\dfrac{1}{\sin^2 x}$	$-\cot x + C$
$\dfrac{1}{\sqrt{1-x^2}}$	$\arcsin x + C$
$\dfrac{-1}{\sqrt{1-x^2}}$	$\arccos x + C$
$\dfrac{1}{1+x^2}$	$\arctan x + C$
$\dfrac{-1}{1+x^2}$	$\arccot x + C$

Hiperbolikus függvények

$f(x)$	$F(x)$
$\sinh x$	$\cosh x + C$
$\cosh x$	$\sinh x + C$
$\dfrac{1}{\cosh^2 x}$	$\tanh x + C$
$\dfrac{1}{\sinh^2 x}$	$\coth x + C$
$\dfrac{1}{\sqrt{x^2+1}}$	$\arcsinh x + C$
$\dfrac{1}{\sqrt{x^2-1}}$	$\arccosh x + C$
$\dfrac{1}{1-x^2}$	$\arctanh x + C$
$\dfrac{1}{x^2-1}$	$\arccoth x + C$

Linearitás teljesülése

$$\begin{gathered} \int \lambda f(x) \dd x = \lambda \int f(x) \dd x \\ \int f(x) + g(x) \dd x = \int f(x) \dd x + \int g(x) \dd x \\ \int_{a}^{b} f(x) \dd x = \int_{a}^{c} f(x) \dd x + \int_{c}^{b} f(x) \dd x \end{gathered}$$

Partiális integrálás

$$\int f'(x) g(x) \dd x = f(x) g(x) - \int f(x) g'(x) \dd x$$

Helyettesítéses integrálás

$$\begin{aligned} \int e^{f(x)} f'(x) \dd x &= e^{f(x)} + C \\ \int \frac{f'(x)}{f(x)} \dd x &= \ln \abs{f(x)} + C \\ \int f^n(x) f'(x) \dd x &= \frac{f^{n+1}(x)}{n+1} + C \\ \int f(g(x)) g'(x) \dd x &= F(g(x)) + C \end{aligned}$$