Sivu:Lukion taulukot.pdf/37

36 MERKINTÖJÄ, MÄÄRITELMIÄ JA KAAVOJA

---

Intergraalifunktioita

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