Lijst van integralen

Uit Wikipedia, de vrije encyclopedie

Het integreren is een basisbewerking uit de analyse. Aangezien het integreren niet aan eenvoudige regels voldoet (zoals het differentiëren), zijn tabellen met veel voorkomende integralen handig.

Hieronder zal een C voor een constante staan, die enkel met bijkomende informatie (beginvoorwaarde / randvoorwaarde) ingevuld kan worden.

[bewerk] Regels bij het integreren

Integraalrekening is lineair:

$\int cf(x)\,{\rm d}x = c\int f(x)\,{\rm d}x$

$\int [f(x) + g(x)]\,{\rm d}x = \int f(x)\,{\rm d}x + \int g(x)\,{\rm d}x$

Partiële integratie

$\int f(x)g(x)\,{\rm d}x = f(x)\int g(x)\,{\rm d}x - \int \left(d[f(x)]\int g(x)\,{\rm d}x\right)\,{\rm d}x$

Bepaalde integralen

$\int _a ^b \frac{dF(x)}{{\rm d}x}\,{\rm d}x = [F(x)] _a ^b = F(b)-F(a)$

Meervoudige integralen

$\int \int F(x,y)\,{\rm d}x\,dy$

Complexe integralen

$\int _{a+ib} ^{c+id} F(x+iy)\,d(x+iy)$

Substitutie

$\int f'(g(x))g'(x)\,{\rm d}x=f(g(x))+c$

[bewerk] Integralen van standaardfuncties

[bewerk] Rationale functies

$\int 1\,{\rm d}x = x + C$

$\int x^n\,{\rm d}x = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ als }n \ne -1$

$\int \frac{1}{x}\,{\rm d}x = \ln{\left|x\right|} + C$

$\int \frac{1}{a^2+x^2}\,{\rm d}x = \frac{1}{a}\arctan \frac{x}{a} + C$

$\int \frac{1}{x\left(a+bx\right)}\,{\rm d}x = \frac{1}{a}\ln\left|\frac{x}{a+bx}\right| + C$

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

$\int \frac{x}{ax^2+bx+c}\,{\rm d}x = \frac{1}{2a}\ln\left|ax^2+bx+x\right|-\frac{b}{2a}\int \frac{1}{ax^2+bx+c}\,{\rm d}x$

[bewerk] Logaritmes

$\int \ln {x}\,{\rm d}x = x \ln {x} - x + C$

$\int \log_b {x}\,{\rm d}x = x\log_b {x} - x\log_b {e} + C$

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

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

[bewerk] Exponentiële functies

$\int e^x\,{\rm d}x = e^x + C$

$\int a^x\,{\rm d}x = \frac{a^x}{\ln{a}} + C$

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

[bewerk] Irrationale functies

$\int {du \over \sqrt{a^2-u^2}} = \arcsin {u \over a} + C$

$\int {-du \over \sqrt{a^2-u^2}} = \arccos {u \over a} + C$

$\int {du \over u\sqrt{u^2-a^2}} = {1 \over a}\mbox{arcsec}\,{|u| \over a} + C$

$\int \sqrt{a^2-x^2}\,{\rm d}x = \frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\frac{x}{a} + C ,(a>0)$

$\int \left(a^2-x^2\right)^{\frac{3}{2}}\,{\rm d}x = \frac{x}{8}\left(5a^2-2x^2\right)\sqrt{a^2-x^2}+\frac{3a^4}{8}\arcsin\frac{x}{a} + C,(a>0)$

$\int \frac{1}{\left(a^2-x^2\right)^{\frac{3}{2}}}\,{\rm d}x = \frac{x}{a^2\sqrt{a^2-x^2}} + C$

$\int x\sqrt{a+bx}\,{\rm d}x = \frac{2\left(3bx-2a\right)\left(a+bx\right)^{\frac{3}{2}}}{15b^2} + C$

$\int \frac{\sqrt{a+bx}}{x}\,{\rm d}x = 2\sqrt{a+bx}+a\int \frac{1}{x\sqrt{a+bx}}\,{\rm d}x$

$\int \frac{x}{\sqrt{a+bx}}\,{\rm d}x = \frac{2\left(bx-2a\right)\sqrt{a+bx}}{3b^2}+C$

$\int \frac{1}{x\sqrt{a+bx}}\,{\rm d}x = \frac{1}{\sqrt{a}}\ln\left|\frac{\sqrt{a+bx}-\sqrt{a}}{\sqrt{a+bx}+\sqrt{a}}\right|+C,(a>0)$

$\int \frac{1}{x\sqrt{a+bx}}\,{\rm d}x = \frac{2}{\sqrt{-a}}\arctan\left|\sqrt{\frac{a+bx}{-a}}\right|+C,(a<0)$

$\int \frac{\sqrt{a^2-x^2}}{x}\,{\rm d}x = \sqrt{a^2-x^2}-a\ln\left|\frac{a+\sqrt{a^2+x^2}}{x}\right|+C$

$\int x\sqrt{a^2-x^2}\,{\rm d}x = -\frac{1}{3}\left(a^2-x^2\right)^{\frac{3}{2}}+C$

$\int x^2\sqrt{a^2-x^2}\,{\rm d}x = \frac{x}{8}\left(2x^2-a^2\right)\sqrt{a^2-x^2}+\frac{a^4}{8}\arcsin\frac{x}{a}+C,(a>0)$

$\int \frac{1}{x\sqrt{a^2-x^2}}\,{\rm d}x = -\frac{1}{a}\ln\left|\frac{a+\sqrt{a^2-x^2}}{x}\right|+C$

$\int \frac{x}{\sqrt{a^2-x^2}}\,{\rm d}x = -\sqrt{a^2-x^2}+C$

$\int \frac{x^2}{\sqrt{a^2-x^2}}\,{\rm d}x = -\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\frac{x}{a}+C,(a>0)$

$\int \frac{\sqrt{x^2+a^2}}{x}\,{\rm d}x = \sqrt{x^2+a^2}-a\ln\left|\frac{a+\sqrt{x^2+a^2}}{x}\right|+C$

$\int \frac{\sqrt{x^2-a^2}}{x}\,{\rm d}x = \sqrt{x^2-a^2}-a\arccos\frac{a}{|x|}+C,(a>0)$

$\int \frac{x^2}{\sqrt{x^2+a^2}}\,{\rm d}x = \frac{x\sqrt{x^2+a^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right)+C$

$\int \frac{1}{x\sqrt{x^2+a^2}}\,{\rm d}x = \frac{1}{a}\ln\left|\frac{x}{a+\sqrt{x^2+a^2}}\right|+C$

$\int \frac{1}{x^2\sqrt{x^2\pm a^2}}\,{\rm d}x = \mp\frac{\sqrt{x^2\pm a^2}}{a^2x}+C$

$\int \frac{1}{\sqrt{x^2\pm a^2}}\,{\rm d}x = \ln\left|\frac{x+\sqrt{x^2\pm a^2}}{a}\right|+C =\operatorname{arcsinh}\frac{x}{a}+C$

$\int \frac{1}{\sqrt{ax^2+bx+c}}\,{\rm d}x = \frac{1}{\sqrt{a}}\ln\left|2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}\right|+C,(a>0)$

$\int \frac{1}{\sqrt{ax^2+bx+c}}\,{\rm d}x = \frac{1}{\sqrt{-a}}\arcsin\frac{-2ax-b}{\sqrt{b^2-4ac}}+C,(a<0)$

$\int \sqrt{ax^2+bx+c}\,{\rm d}x = \frac{2ax+b}{4a}\sqrt{ax^2+bx+c}+\frac{4ac-b^2}{8a}\int \frac{1}{\sqrt{ax^2+bx+c}}\,{\rm d}x$

$\int \frac{x}{\sqrt{ax^2+bx+c}}\,{\rm d}x = \frac{\sqrt{ax^2+bx+c}}{a}-\frac{b}{2a}\int \frac{1}{\sqrt{ax^2+bx+c}}\,{\rm d}x$

$\int \frac{1}{x\sqrt{ax^2+bx+c}}\,{\rm d}x = \frac{-1}{\sqrt{c}}\ln\left|\frac{2\sqrt{c}\sqrt{ax^2+bx+c}+bx+2c}{x}\right|+C,(c>0)$

$\int \frac{1}{x\sqrt{ax^2+bx+c}}\,{\rm d}x = \frac{1}{\sqrt{-c}}\arcsin\frac{bx+2c}{|x|\sqrt{b^2-4ac}}+C,(c<0)$

$\int x^3\sqrt{x^2+a^2}\,{\rm d}x = \left(\frac{1}{5}x^2-\frac{2}{15}a^2\right)\sqrt{\left(x^2+x^2\right)^3}+C$

$\int \frac{\sqrt{x^2\pm a^2}}{x^4}\,{\rm d}x = \frac{\mp \sqrt{\left(x^2+a^2\right)^3}}{3a^2x^3}+C$

[bewerk] Trigonometrische functies

$\int \sin{x}\, {\rm d}x = -\cos{x} + C$

$\int \cos{x}\, {\rm d}x = \sin{x} + C$

$\int \tan{x} \, {\rm d}x = -\ln{\left| \cos {x} \right|} + C$

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

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

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

$\int \frac{1}{\sin x}\,{\rm d}x = \ln\left|\tan\tfrac12 x\right|+C = \ln\left|\frac{1}{\sin x}-\cot x\right|+C$

$\int \frac{1}{\cos x}\,{\rm d}x = \ln\left|\tan\tfrac12 x+\tfrac14\pi\right|+C = \ln\left|\frac{1}{\cos x}+\tan x\right|+C$

$\int \arcsin{\frac{x}{a}}\, {\rm d}x = x\arcsin{\frac{x}{a}}+\sqrt{a^2-x^2} + C,(a>0)$

$\int \arccos{\frac{x}{a}}\, {\rm d}x = x\arccos{\frac{x}{a}}-\sqrt{a^2-x^2} + C,(a>0)$

$\int \arctan{\frac{x}{a}}\, {\rm d}x = x\arctan{\frac{x}{a}}-\frac{a}{2}\ln{a^2+x^2} + C ,(a>0)$

$\int \frac{1}{\cos^2 x} \, {\rm d}x = \int \sec^2 x \, {\rm d}x = \tan x + C$

$\int \frac{1}{\sin^2 x} \, {\rm d}x = \int \csc^2 x \, {\rm d}x = -\cot x + C$

$\int \sec{x} \, \tan{x} \, {\rm d}x = \sec{x} + C$

$\int \csc{x} \, \cot{x} \, {\rm d}x = - \csc{x} + C$

$\int \sin^2 x \, {\rm d}x = \tfrac12(x - \sin x \cos x) + C$

$\int \cos^2 x \, {\rm d}x = \tfrac12(x + \sin x \cos x) + C$

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

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

$\int \tan^n x \, {\rm d}x = \frac{\tan^{n-1}x}{n-1}-\int\tan^{n-2}x \, {\rm d}x ,(n\neq1)$

$\int \cot^n x \, {\rm d}x = -\frac{\cot^{n-1}x}{n-1}-\int \cot^{n-2}x \, {\rm d}x ,(n\neq1)$

$\int \sec^n x \, {\rm d}x = \frac{\tan x\sec^{n-2}x}{n-1}+\frac{n-2}{n-1}\int \sec^{n-2}x \, {\rm d}x,(n\neq1)$

$\int \csc^n x \, {\rm d}x = -\frac{\cot x\csc^{n-2}x}{n-1}+\frac{n-2}{n-1}\int \csc^{n-2}x \, {\rm d}x,(n\neq1)$

$\int \sin ax\sin bx\,{\rm d}x = \frac{\sin(a-b)x}{2(a-b)}-\frac{\sin(a+b)x}{2(a+b)}+C,(a^2\neq b^2)$

$\int \sin ax\cos bx\,{\rm d}x = -\frac{\cos(a-b)x}{2(a-b)}-\frac{\cos(a+b)x}{2(a+b)}+C,(a^2\neq b^2)$

$\int \cos ax\cos bx\,{\rm d}x = \frac{\sin(a-b)x}{2(a-b)}-\frac{\sin(a+b)x}{2(a+b)}+C,(a^2\neq b^2)$

$\int \sec x\tan x\,{\rm d}x = \sec x+C$

$\int \csc x\cot x\,{\rm d}x = -\csc x+C$

$\int \cos^mx\sin^nx\,{\rm d}x = \frac{\cos^{m-1}x\sin^{n+1}x}{m+n}+\frac{m-1}{m+n}\int \cos^{m-2}x\sin^nx\,{\rm d}x$

$=-\frac{\sin^{n-1}x\cos^{m+1}x}{m+n}+\frac{n-1}{m+n}\int \cos^mx\sin^{n-2}x\,{\rm d}x$

$\int x^n\sin ax\,{\rm d}x = -\frac{1}{a}x^n\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\,{\rm d}x$

$\int x^n\cos ax\,{\rm d}x = \frac{1}{a}x^n\sin ax -\frac{n}{a}\int x^{n-1}\sin ax\,{\rm d}x$

$\int e^{ax}\sin bx\,{\rm d}x = \frac{e^{ax}\left(a\sin bx-b\cos bx\right)}{a^2+b^2}+C$

$\int e^{ax}\cos bx\,{\rm d}x = \frac{e^{ax}\left(b\sin bx+a\cos bx\right)}{a^2+b^2}+C$

[bewerk] Hyperbolische functies

$\int \sinh x \, {\rm d}x = \cosh x + C$

$\int \cosh x \, {\rm d}x = \sinh x + C$

$\int \tanh x \, {\rm d}x = \ln |\cosh x| + C$

$\int \mbox{csch}\,x \, {\rm d}x = \ln\left| \tanh {x \over2}\right| + C$

$\int \mbox{sech}\,x \, {\rm d}x = \arctan(\sinh x) + C$

$\int \coth x \, {\rm d}x = \ln|\sinh x| + C$

$\int \sinh^2 x \, {\rm d}x = \frac{1}{4}\sinh 2x-\frac{1}{2}x + C$

$\int \cosh^2 x \, {\rm d}x = \frac{1}{4}\sinh 2x+\frac{1}{2}x + C$

$\int \mbox{sech}^2 x \, {\rm d}x = \tanh x + C$

$\int \sinh^{-1}\frac{x}{a} \, {\rm d}x = x\sinh^{-1}\frac{x}{a}-\sqrt{x^2+a^2} + C$

$\int \cosh^{-1}\frac{x}{a} \, {\rm d}x = x\cosh^{-1}\frac{x}{a}-\sqrt{x^2-a^2} + C \left(\cosh^{-1}\frac{x}{a}>0,a>0\right)$

$\int \cosh^{-1}\frac{x}{a} \, {\rm d}x = x\cosh^{-1}\frac{x}{a}+\sqrt{x^2-a^2} + C \left(\cosh^{-1}\frac{x}{a}<0,a>0\right)$

$\int \tanh^{-1}\frac{x}{a} \, {\rm d}x = x\tanh^{-1}\frac{x}{a}+\frac{a}{2}\ln\left|a^2-x^2\right|+C$

$\int \mbox{sech}x\tanh x\,{\rm d}x = -\mbox{sech}x+C$

$\int \mbox{csch}x\coth c\,{\rm d}x = -\mbox{csch}x+C$

[bewerk] Oneigenlijke integralen

Voor sommige functies kan de primitieve functie niet gevonden worden, hoewel de waarde ervan wel bekend is (en op andere manieren berekend)

$\int_0^\infty{\sqrt{x}\,e^{-x}\,{\rm d}x} = \frac{1}{2}\sqrt \pi$

$\int_0^\infty{e^{-x^2}\,{\rm d}x} = \frac{1}{2}\sqrt \pi$

$\int_0^\infty{\frac{x}{e^x-1}\,{\rm d}x} = \frac{\pi^2}{6}$

$\int_0^\infty{\frac{x^3}{e^x-1}\,{\rm d}x} = \frac{\pi^4}{15}$

$\int_0^\infty\frac{\sin(x)}{x}\,{\rm d}x=\frac{\pi}{2}$

$\int_0^\infty x^{z-1}\,e^{-x}\,{\rm d}x = \Gamma(z)$ (met

Γ(z)

de Gammafunctie.)

Categorieën: Integraalrekening | Wiskundelijsten

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