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# Lukion taulukot/Integraalifunktioita

#### Intergraalifunktioita

${\displaystyle f(x)}$ ${\displaystyle F(x)=\textstyle \int f(x)\mathrm {d} x+C}$
${\displaystyle k}$ ${\displaystyle kx+C}$
${\displaystyle x^{n}(n\neq -1)}$ ${\displaystyle {\frac {1}{n+1}}x^{n+1}+C}$
${\displaystyle {\frac {1}{x}}(x\neq 0)}$ ${\displaystyle \ln |x|+C}$
${\displaystyle e^{x}}$ ${\displaystyle e^{x}+C}$
${\displaystyle a^{x}(a>0,a\neq 1)}$ ${\displaystyle {\frac {1}{\ln a}}a^{x}+C}$
${\displaystyle {\frac {1}{x-a}}}$ ${\displaystyle \ln |x-a|+C}$
${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}$ ${\displaystyle \arcsin x+C}$
${\displaystyle {\frac {1}{x^{2}+a^{2}}}}$ ${\displaystyle {\frac {1}{a}}\arctan {\frac {x}{a}}+C}$
${\displaystyle {\frac {x}{x^{2}+a^{2}}}}$ ${\displaystyle {\frac {1}{2}}\ln(x^{2}+a^{2})+C}$
${\displaystyle \sin x}$ ${\displaystyle -\cos x+C}$
${\displaystyle \sin ^{2}x}$ ${\displaystyle {\frac {1}{2}}(x-\sin x\cos x)+C}$
${\displaystyle \cos x}$ ${\displaystyle \sin x+C}$
${\displaystyle \cos ^{2}x}$ ${\displaystyle {\frac {1}{2}}(x+\sin x\cos x)+C}$
${\displaystyle \tan x}$ ${\displaystyle -\ln |\cos x|+C}$
${\displaystyle \cot x}$ ${\displaystyle \ln |\sin x|+C}$
${\displaystyle {\frac {1}{\sin ^{2}x}}}$ ${\displaystyle -\cot x+C}$
${\displaystyle {\frac {1}{\cos ^{2}x}}}$ ${\displaystyle \tan x+C}$
${\displaystyle f(x)^{n}\cdot f'(x)(n\in \mathbb {R} ,n\neq -1)}$ ${\displaystyle {\frac {1}{n+1}}f(x)^{n+1}+C}$
${\displaystyle {\frac {f'(x)}{f(x)}}}$ ${\displaystyle \ln |f(x)|+C}$
${\displaystyle \mathrm {e} ^{f(x)}f'(x)}$ ${\displaystyle \mathrm {e} ^{f(x)}+C}$
${\displaystyle \ln |x|\;(x\neq 0)}$ ${\displaystyle x\ln |x|-x+C}$