Luettelo integraaleista

Tässä on luettelo tavallisimpien matemaattisten operaattoreiden sekä funktioiden integraaleista.

Rationaalifunktiot

$\int kdx=kx+C$

$\int x^{a}dx=\frac{x^{a+1}}{a+1}+C(a\ne -1\text{)}$

$\int (ax+b{)}^{n}dx=\frac{(ax+b{)}^{n+1}}{a(n+1)}+C(n\ne -1\text{)}$

$\int \frac{1}{x}dx=\ln \left|x\right|+C$

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

Eksponenttifunktiot

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

$\int f^{\prime }(x)e^{f(x)}dx=e^{f(x)}+C$

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

Logaritmifunktiot

$\int \ln xdx=x\ln x-x+C$

$\int {\log }_{a}xdx=x{\log }_{a}x-\frac{x}{\ln a}+C$

Trigonometriset funktiot

$\int \sin xdx=-\cos x+C$

$\int \cos xdx=\sin x+C$

$\int \tan xdx=-\ln |\cos x|+C=\ln |\sec x|+C$

$\int \cot xdx=\ln |\sin x|+C$

$\int \sec xdx=\ln |\sec x+\tan x|+C$

$\int \csc xdx=\ln |\csc x-\cot x|+C$

$\int {\sec }^{2}xdx=\tan x+C$

$\int {\csc }^{2}xdx=-\cot x+C$

$\int \sec x\tan xdx=\sec x+C$

$\int \csc x\cot xdx=-\csc x+C$

$\int {\sin }^{2}xdx=\frac{1}{2}(x-\frac{\sin 2x}{2})+C=\frac{1}{2}(x-\sin x\cos x)+C$

$\int {\cos }^{2}xdx=\frac{1}{2}(x+\frac{\sin 2x}{2})+C=\frac{1}{2}(x+\sin x\cos x)+C$

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

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

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

Arkusfunktiot

$\int \arcsin xdx=x\arcsin x+\sqrt{1-x^2}+C,|x|\le +1$

$\int \arccos xdx=x\arccos x-\sqrt{1-x^2}+C,|x|\le +1$

$\int \arctan xdx=x\arctan x-\frac{1}{2}\ln |1+x^2|+C,$

$\int \mathrm{arccot}xdx=x\mathrm{arccot}x+\frac{1}{2}\ln |1+x^2|+C,$

$\int \mathrm{arcsec}xdx=x\mathrm{arcsec}x-\ln |x(1+\sqrt{1-x^{-2}})|+C,|x|\ge +1$

$\int \mathrm{arccsc}xdx=x\mathrm{arccsc}x+\ln |x(1+\sqrt{1-x^{-2}})|+C,|x|\ge +1$

Hyperboliset funktiot

$\int \sinh xdx=\cosh x+C$

$\int \cosh xdx=\sinh x+C$

$\int \tanh xdx=\ln \cosh x+C$

$\int \coth xdx=\ln |\sinh x|+C,x\ne 0$

$\int \mathrm{sech}xdx=\arctan (\sinh x)+C$

$\int \mathrm{csch}xdx=\ln |\tanh \frac{x}{2}|+C,x\ne 0$

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

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

Lähteet

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Kirjallisuutta

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