Lijst van integralen

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Integreren is een basisbewerking uit de analyse. Aangezien integreren niet, zoals bij differentiëren, door eenvoudige regels plaatsvindt, zijn tabellen met veel voorkomende integralen een handig hulpmiddel.

In de onderstaande lijst van integralen is C steeds een integratieconstante die enkel met bijkomende informatie, beginvoorwaarde of randvoorwaarde, bepaald kan worden. Voor meer uitgebreide lijsten, zie onderstaande artikelen:


Rekenregels bij het integreren[bewerken]

\int cf(x)\,\operatorname{d}x = c\int f(x)\,dx
\int (f(x) + g(x))\,\operatorname{d}x = \int f(x)\,\operatorname{d}x + \int g(x)\,\operatorname{d}x
\int f(x)g'(x)\,\operatorname{d}x = f(x)g(x) - \int f'(x)g(x)\,\operatorname{d}x
  • Bepaalde integraal
\int _a ^b \frac{\operatorname{d}F(x)}{\operatorname{d}x}\,\operatorname{d}x = [F(x)] _a ^b = F(b)-F(a)
  • Meervoudige integraal als herhaalde integraal
\iint f(x,y)\,\operatorname{d}x\operatorname{d}y= \int \left(\int f(x,y)\,\operatorname{d}x\right)\operatorname{d}y
\int f(g(t))\,g'(t)\,\operatorname{d}t=\int f(x)\,\operatorname{d}x

Integralen van standaardfuncties[bewerken]

Rationale functies[bewerken]

\int 1\,{\rm d}x = x + C
\int x^n\,{\rm d}x =  \frac{x^{n+1}}{n+1} + C\qquad\mbox{ als }n \ne -1
\int \frac{1}{x}\,{\rm d}x = \ln{\left|x\right|} + C
\int \frac{1}{a^2+x^2}\,{\rm d}x = \frac{1}{a}\arctan \frac{x}{a} + C
\int \frac{1}{x\left(a+bx\right)}\,{\rm d}x = \frac{1}{a}\ln\left|\frac{x}{a+bx}\right| + C
\int \frac{1}{ax^2+bx+c}\,{\rm d}x = \left\{ 
 \begin{matrix} 
  \cfrac{1}{\sqrt{b^2-4ac}}\ln\left|\cfrac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right| + C
  &
  \mbox{als}\ b^2>4ac
  \\
  \cfrac{2}{\sqrt{4ac-b^2}}\arctan\cfrac{2ax+b}{\sqrt{4ac-b^2}} + C
  &
  \mbox{als}\ b^2<4ac
 \end{matrix}\right.
\int \frac{x}{ax^2+bx+c}\,{\rm d}x = \frac{1}{2a}\ln\left|ax^2+bx+c\right|-\frac{b}{2a}\int \frac{1}{ax^2+bx+c}\,{\rm d}x

Logaritmes[bewerken]

\int \ln {x}\,{\rm d}x = x \ln {x} - x + C
\int \log_b {x}\,{\rm d}x = x\log_b {x} - x\log_b {e} + C
\int x^n\ln ax\,{\rm d}x = x^{n+1}\left(\frac{\ln ax}{n+1}-\frac{1}{(n+1)^2}\right)+C
\int x^n\left(\ln ax\right)^m\,{\rm d}x = \frac{x^{n+1}}{n+1}\left(\ln ax\right)^m-\frac{m}{n+1}\int x^n\left(\ln ax\right)^{m-1}\,{\rm d}x

Exponentiële functies[bewerken]

\int e^x\,{\rm d}x = e^x + C
\int a^x\,{\rm d}x = \frac{a^x}{\ln{a}} + C
\int e^{ax}\,{\rm d}x = \frac{e^{ax}}{a}+ C
\int x^ne^{ax}\,{\rm d}x = \frac{x^ne^{ax}}{a}-\frac{n}{a}\int x^{n-1}e^{ax}\,{\rm d}x

Irrationale functies[bewerken]

\int {\operatorname{d}u \over \sqrt{a^2-u^2}} = \arcsin {u \over a} + C
\int {-\operatorname{d}u \over \sqrt{a^2-u^2}} = \arccos {u \over a} + C
\int {\operatorname{d}u \over u\sqrt{u^2-a^2}} = {1 \over a}\mbox{arcsec}\,{|u| \over a} + C
\int \sqrt{a^2-x^2}\,{\rm d}x = \frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\frac{x}{a} + C ,(a>0)
\int \left(a^2-x^2\right)^{\frac{3}{2}}\,{\rm d}x = \frac{x}{8}\left(5a^2-2x^2\right)\sqrt{a^2-x^2}+\frac{3a^4}{8}\arcsin\frac{x}{a} + C,(a>0)
\int \frac{1}{\left(a^2-x^2\right)^{\frac{3}{2}}}\,{\rm d}x = \frac{x}{a^2\sqrt{a^2-x^2}} + C
\int x\sqrt{a+bx}\,{\rm d}x = \frac{2\left(3bx-2a\right)\left(a+bx\right)^{\frac{3}{2}}}{15b^2} + C
\int \frac{\sqrt{a+bx}}{x}\,{\rm d}x = 2\sqrt{a+bx}+a\int \frac{1}{x\sqrt{a+bx}}\,{\rm d}x
\int \frac{x}{\sqrt{a+bx}}\,{\rm d}x = \frac{2\left(bx-2a\right)\sqrt{a+bx}}{3b^2}+C
\int \frac{1}{x\sqrt{a+bx}}\,{\rm d}x = \frac{1}{\sqrt{a}}\ln\left|\frac{\sqrt{a+bx}-\sqrt{a}}{\sqrt{a+bx}+\sqrt{a}}\right|+C,(a>0)
\int \frac{1}{x\sqrt{a+bx}}\,{\rm d}x = \frac{2}{\sqrt{-a}}\arctan\left|\sqrt{\frac{a+bx}{-a}}\right|+C,(a<0)
\int \frac{\sqrt{a^2-x^2}}{x}\,{\rm d}x = \sqrt{a^2-x^2}-a\ln\left|\frac{a+\sqrt{a^2+x^2}}{x}\right|+C
\int x\sqrt{a^2-x^2}\,{\rm d}x = -\frac{1}{3}\left(a^2-x^2\right)^{\frac{3}{2}}+C
\int x^2\sqrt{a^2-x^2}\,{\rm d}x = \frac{x}{8}\left(2x^2-a^2\right)\sqrt{a^2-x^2}+\frac{a^4}{8}\arcsin\frac{x}{a}+C,(a>0)
\int \frac{1}{x\sqrt{a^2-x^2}}\,{\rm d}x = -\frac{1}{a}\ln\left|\frac{a+\sqrt{a^2-x^2}}{x}\right|+C
\int \frac{x}{\sqrt{a^2-x^2}}\,{\rm d}x = -\sqrt{a^2-x^2}+C
\int \frac{x^2}{\sqrt{a^2-x^2}}\,{\rm d}x = -\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\frac{x}{a}+C,(a>0)
\int \frac{\sqrt{x^2+a^2}}{x}\,{\rm d}x = \sqrt{x^2+a^2}-a\ln\left|\frac{a+\sqrt{x^2+a^2}}{x}\right|+C
\int \frac{\sqrt{x^2-a^2}}{x}\,{\rm d}x = \sqrt{x^2-a^2}-a\arccos\frac{a}{|x|}+C,(a>0)
\int \frac{x^2}{\sqrt{x^2+a^2}}\,{\rm d}x = \frac{x\sqrt{x^2+a^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right)+C
\int \frac{1}{x\sqrt{x^2+a^2}}\,{\rm d}x = \frac{1}{a}\ln\left|\frac{x}{a+\sqrt{x^2+a^2}}\right|+C
\int \frac{1}{x^2\sqrt{x^2\pm a^2}}\,{\rm d}x = \mp\frac{\sqrt{x^2\pm a^2}}{a^2x}+C
\int \frac{1}{\sqrt{x^2\pm a^2}}\,{\rm d}x = \ln\left|\frac{x+\sqrt{x^2\pm a^2}}{a}\right|+C =\operatorname{arcsinh}\frac{x}{a}+C
\int \frac{1}{\sqrt{ax^2+bx+c}}\,{\rm d}x = \frac{1}{\sqrt{a}}\ln\left|2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}\right|+C,(a>0)
\int \frac{1}{\sqrt{ax^2+bx+c}}\,{\rm d}x = \frac{1}{\sqrt{-a}}\arcsin\frac{-2ax-b}{\sqrt{b^2-4ac}}+C,(a<0)
\int \sqrt{ax^2+bx+c}\,{\rm d}x = \frac{2ax+b}{4a}\sqrt{ax^2+bx+c}+\frac{4ac-b^2}{8a}\int \frac{1}{\sqrt{ax^2+bx+c}}\,{\rm d}x
\int \frac{x}{\sqrt{ax^2+bx+c}}\,{\rm d}x = \frac{\sqrt{ax^2+bx+c}}{a}-\frac{b}{2a}\int \frac{1}{\sqrt{ax^2+bx+c}}\,{\rm d}x
\int \frac{1}{x\sqrt{ax^2+bx+c}}\,{\rm d}x = \frac{-1}{\sqrt{c}}\ln\left|\frac{2\sqrt{c}\sqrt{ax^2+bx+c}+bx+2c}{x}\right|+C,(c>0)
\int \frac{1}{x\sqrt{ax^2+bx+c}}\,{\rm d}x = \frac{1}{\sqrt{-c}}\arcsin\frac{bx+2c}{|x|\sqrt{b^2-4ac}}+C,(c<0)
\int x^3\sqrt{x^2+a^2}\,{\rm d}x = \left(\frac{1}{5}x^2-\frac{2}{15}a^2\right)\sqrt{\left(x^2+x^2\right)^3}+C
\int \frac{\sqrt{x^2\pm a^2}}{x^4}\,{\rm d}x = \frac{\mp \sqrt{\left(x^2+a^2\right)^3}}{3a^2x^3}+C

Goniometrische functies[bewerken]

\int \sin{x}\, {\rm d}x = -\cos{x} + C
\int \cos{x}\, {\rm d}x = \sin{x} + C
\int \tan{x} \, {\rm d}x = -\ln{\left| \cos {x} \right|} + C
\int \cot{x} \, {\rm d}x = \ln{\left| \sin{x} \right|} + C
\int \sec{x} \, {\rm d}x = \ln{\left| \sec{x} + \tan{x}\right|} + C
\int \csc{x} \, {\rm d}x = \ln{\left| \csc{x} - \cot{x}\right|} + C
\int \frac{1}{\sin x}\,{\rm d}x = \ln\left|\tan\tfrac12 x\right|+C = \ln\left|\frac{1}{\sin x}-\cot x\right|+C
\int \frac{1}{\cos x}\,{\rm d}x = \ln\left|\tan\tfrac12 x+\tfrac14\pi\right|+C = \ln\left|\frac{1}{\cos x}+\tan x\right|+C
\int \arcsin{\frac{x}{a}}\, {\rm d}x = x\arcsin{\frac{x}{a}}+\sqrt{a^2-x^2} + C,(a>0)
\int \arccos{\frac{x}{a}}\, {\rm d}x = x\arccos{\frac{x}{a}}-\sqrt{a^2-x^2} + C,(a>0)
\int \arctan{\frac{x}{a}}\, {\rm d}x = x\arctan{\frac{x}{a}}-\frac{a}{2}\ln\left(a^2+x^2\right)+C ,(a>0)
\int \frac{1}{\cos^2 x} \, {\rm d}x = \int \sec^2 x \, {\rm d}x = \tan x + C
\int \frac{1}{\sin^2 x} \, {\rm d}x = \int \csc^2 x \, {\rm d}x = -\cot x + C
\int \sec{x} \, \tan{x} \, {\rm d}x = \sec{x} + C
\int \csc{x} \, \cot{x} \, {\rm d}x = - \csc{x} + C
\int \sin^2 x \, {\rm d}x = \tfrac12(x - \sin x \cos x) + C
\int \cos^2 x \, {\rm d}x = \tfrac12(x + \sin x \cos x) + C
\int \sin^n x \, {\rm d}x = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, {\rm d}x
\int \cos^n x \, {\rm d}x = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, {\rm d}x
\int \tan^n x \, {\rm d}x = \frac{\tan^{n-1}x}{n-1}-\int\tan^{n-2}x \, {\rm d}x ,(n\neq1)
\int \cot^n x \, {\rm d}x = -\frac{\cot^{n-1}x}{n-1}-\int \cot^{n-2}x \, {\rm d}x ,(n\neq1)
\int \sec^n x \, {\rm d}x = \frac{\tan x\sec^{n-2}x}{n-1}+\frac{n-2}{n-1}\int \sec^{n-2}x \, {\rm d}x,(n\neq1)
\int \csc^n x \, {\rm d}x = -\frac{\cot x\csc^{n-2}x}{n-1}+\frac{n-2}{n-1}\int \csc^{n-2}x \, {\rm d}x,(n\neq1)
\int \sin ax\sin bx\,{\rm d}x = \frac{\sin(a-b)x}{2(a-b)}-\frac{\sin(a+b)x}{2(a+b)}+C,(a^2\neq b^2)
\int \sin ax\cos bx\,{\rm d}x = -\frac{\cos(a-b)x}{2(a-b)}-\frac{\cos(a+b)x}{2(a+b)}+C,(a^2\neq b^2)
\int \cos ax\cos bx\,{\rm d}x = \frac{\sin(a-b)x}{2(a-b)}-\frac{\sin(a+b)x}{2(a+b)}+C,(a^2\neq b^2)
\int \sec x\tan x\,{\rm d}x = \sec x+C
\int \csc x\cot x\,{\rm d}x = -\csc x+C
\int \cos^mx\sin^nx\,{\rm d}x = \frac{\cos^{m-1}x\sin^{n+1}x}{m+n}+\frac{m-1}{m+n}\int \cos^{m-2}x\sin^nx\,{\rm d}x
=-\frac{\sin^{n-1}x\cos^{m+1}x}{m+n}+\frac{n-1}{m+n}\int \cos^mx\sin^{n-2}x\,{\rm d}x
\int x^n\sin ax\,{\rm d}x = -\frac{1}{a}x^n\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\,{\rm d}x
\int x^n\cos ax\,{\rm d}x = \frac{1}{a}x^n\sin ax -\frac{n}{a}\int x^{n-1}\sin ax\,{\rm d}x
\int e^{ax}\sin bx\,{\rm d}x = \frac{e^{ax}\left(a\sin bx-b\cos bx\right)}{a^2+b^2}+C
\int e^{ax}\cos bx\,{\rm d}x = \frac{e^{ax}\left(b\sin bx+a\cos bx\right)}{a^2+b^2}+C

Hyperbolische functies[bewerken]

\int \sinh x \, {\rm d}x = \cosh x + C
\int \cosh x \, {\rm d}x = \sinh x + C
\int \tanh x \, {\rm d}x = \ln |\cosh x| + C
\int \mbox{csch}\,x \, {\rm d}x = \ln\left| \tanh {x \over2}\right| + C
\int \mbox{sech}\,x \, {\rm d}x = \arctan(\sinh x) + C
\int \coth x \, {\rm d}x = \ln|\sinh x| + C
\int \sinh^2 x \, {\rm d}x = \frac{1}{4}\sinh 2x-\frac{1}{2}x + C
\int \cosh^2 x \, {\rm d}x = \frac{1}{4}\sinh 2x+\frac{1}{2}x + C
\int \mbox{sech}^2 x \, {\rm d}x = \tanh x + C
\int \sinh^{-1}\frac{x}{a} \, {\rm d}x = x\sinh^{-1}\frac{x}{a}-\sqrt{x^2+a^2} + C
\int \cosh^{-1}\frac{x}{a} \, {\rm d}x = x\cosh^{-1}\frac{x}{a}-\sqrt{x^2-a^2} + C \left(\cosh^{-1}\frac{x}{a}>0,a>0\right)
\int \cosh^{-1}\frac{x}{a} \, {\rm d}x = x\cosh^{-1}\frac{x}{a}+\sqrt{x^2-a^2} + C \left(\cosh^{-1}\frac{x}{a}<0,a>0\right)
\int \tanh^{-1}\frac{x}{a} \, {\rm d}x = x\tanh^{-1}\frac{x}{a}+\frac{a}{2}\ln\left|a^2-x^2\right|+C
\int \mbox{sech}x\tanh x\,{\rm d}x = -\mbox{sech}x+C
\int \mbox{csch}x\coth x\,{\rm d}x = -\mbox{csch}x+C

Oneigenlijke integralen[bewerken]

Voor sommige functies kan de primitieve functie niet in gesloten vorm gevonden worden, maar is de waarde van de integraal over een oneindig integratiegebied wel bekend, omdat die op een andere manier berekend kan worden.

\int_0^\infty{\sqrt{x}\,e^{-x}\,{\rm d}x} = \frac{1}{2}\sqrt \pi
\int_0^\infty{e^{-x^2}\,{\rm d}x} = \frac{1}{2}\sqrt \pi
\int_0^\infty{\frac{x}{e^x-1}\,{\rm d}x} = \frac{\pi^2}{6}
\int_0^\infty{\frac{x^3}{e^x-1}\,{\rm d}x} = \frac{\pi^4}{15}
\int_0^\infty\frac{\sin(x)}{x}\,{\rm d}x=\frac{\pi}{2}