Luettelo integraaleista

Tässä on luettelo tavallisimpien matemaattisten operaattoreiden sekä funktioiden integraaleista.

Rationaalifunktiot

${\displaystyle \int k\,dx=kx+C}$

${\displaystyle \int x^{a}\,dx={\frac {x^{a+1}}{a+1}}+C\qquad \ (a\neq -1{\text{)}}\,\!}$

${\displaystyle \int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad \ (n\neq -1{\text{)}}\,\!}$

${\displaystyle \int {1 \over x}\,dx=\ln \left|x\right|+C}$

${\displaystyle \int {\frac {c}{ax+b}}\,dx={\frac {c}{a}}\ln \left|ax+b\right|+C}$

Eksponenttifunktiot

${\displaystyle \int e^{ax}\,dx={\frac {1}{a}}e^{ax}+C}$

${\displaystyle \int f'(x)e^{f(x)}\,dx=e^{f(x)}+C}$

${\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln a}}+C}$

Logaritmifunktiot

${\displaystyle \int \ln x\,dx=x\ln x-x+C}$

${\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C}$

Trigonometriset funktiot

${\displaystyle \int \sin {x}\,dx=-\cos {x}+C}$

${\displaystyle \int \cos {x}\,dx=\sin {x}+C}$

${\displaystyle \int \tan {x}\,dx=-\ln {\left|\cos {x}\right|}+C=\ln {\left|\sec {x}\right|}+C}$

${\displaystyle \int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C}$

${\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}$

${\displaystyle \int \csc {x}\,dx=\ln {\left|\csc {x}-\cot {x}\right|}+C}$

${\displaystyle \int \sec ^{2}x\,dx=\tan x+C}$

${\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}$

${\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}$

${\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}$

${\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}\left(x-{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x-\sin x\cos x)+C}$

${\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}\left(x+{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x+\sin x\cos x)+C}$

${\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C}$

${\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}$

${\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}$

Arkusfunktiot

${\displaystyle \int \arcsin {x}\,dx=x\arcsin {x}+{\sqrt {1-x^{2}}}+C,\vert x\vert \leq +1}$

${\displaystyle \int \arccos {x}\,dx=x\arccos {x}-{\sqrt {1-x^{2}}}+C,\vert x\vert \leq +1}$

${\displaystyle \int \arctan {x}\,dx=x\arctan {x}-{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,}$

${\displaystyle \int \operatorname {arccot} {x}\,dx=x\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,}$

${\displaystyle \int \operatorname {arcsec} {x}\,dx=x\operatorname {arcsec} {x}-\ln \vert x\,(1+{\sqrt {1-x^{-2}}}\,)\vert +C,\vert x\vert \geq +1}$

${\displaystyle \int \operatorname {arccsc} {x}\,dx=x\operatorname {arccsc} {x}+\ln \vert x\,(1+{\sqrt {1-x^{-2}}}\,)\vert +C,\vert x\vert \geq +1}$

Hyperboliset funktiot

${\displaystyle \int \sinh x\,dx=\cosh x+C}$

${\displaystyle \int \cosh x\,dx=\sinh x+C}$

${\displaystyle \int \tanh x\,dx=\ln \cosh x+C}$

${\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,x\neq 0}$

${\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}$

${\displaystyle \int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C,x\neq 0}$

${\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C\,}$

${\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {2}{a}}\arctan e^{ax}+C\,}$

Lähteet

• Valtanen, Esko: Matemaattisia kaavoja ja taulukoita, s. 78–80. , 2013. ISBN 978-952-9867-37-0.

Kirjallisuutta

• Spiegel, Murray M.: Mathematical Handbook of Formulas and Tables. Shaum's Outline Series. McGraw-Hill, 1992. ISBN 0-07-060224-7.