# Zestawienie pochodnych i całek

W tabeli pominięto stałe całkowania oraz założenia, zaś $a \in \mathbb{R}$.

$\mathbf{f'(x)}$ $\mathbf{f(x)}$ $\mathbf{\int{f(x)dx} - C}$
$0$ $a$ $ax$
$1$ $x$ $\frac{x^2}{2}$
$a$ $ax$ $\frac{ax^2}{2}$
$ax^{a-1}$ $x^a$ $\frac{x^{a+1}}{a+1}$
$-\frac{1}{x^2}$ $\frac{1}{x}\mathbf{ = x^{-1}}$ $\ln|x|$
$\frac{1}{2\sqrt{x}}$ $\sqrt{x}$ $\frac{2x^{3/2}}{3}$
$a^x\ln{a}$ $a^x$ $\frac{a^x}{\ln|a|}$
$e^x$ $e^x$ $e^x$
$\frac{1}{x}$ $\ln{x}$ $x\ln{x}-x$
$\frac{1}{x\ln{a}}$ $\log_{a}{x}$ $\frac{x\ln{x}-x}{\ln{a}}$
$\cos{x}$ $\sin{x}$ $-\cos{x}$
$-\sin{x}$ $\cos{x}$ $\sin{x}$
$\frac{1}{\cos^2 x}$ $\DeclareMathOperator{\tg}{tg}\tg x$ $-\ln|\cos{x}|$
$-\frac{1}{\sin^2 x}$ $\DeclareMathOperator{\ctg}{ctg}\ctg x$ $\ln|\sin x|$
$\frac{1}{\sqrt{1-x^2}}$ $\arcsin x$ $\color{gray}{x\arcsin{x}+\sqrt{1-x^2}}$
$-\frac{1}{\sqrt{1-x^2}}$ $\arccos x$ $\color{gray}{x\arccos{x}-\sqrt{1-x^2}}$
$\frac{1}{1+x^2}$ $\DeclareMathOperator{\arctg}{arctg}\arctg x$ $\color{gray}{x\arctg x - \frac{1}{2}\ln({x^2+1})}$
$-\frac{1}{1+x^2}$ $\DeclareMathOperator{\arcctg}{arcctg}\arcctg x$ $\color{gray}{x\arcctg{x} + \frac{1}{2}\ln(x^2+1)}$