Lijst van integralen van irrationale functies

Dit artikel bevat een lijst van integralen van irrationale functies. Integralen zijn het onderwerp van studie van de integraalrekening. De integralen in de lijst hieronder zijn veel voorkomende integralen van een functie onder de wortel. Er wordt van alle integralen de primitieve functie gegeven, maar de integratieconstante is in de uitkomst steeds weggelaten.

Integralen met $r=\sqrt{x^2+a^2}$

$\int r\mathrm{d}x=\frac{1}{2}(xr+a^2\ln (x+r))$

$\int r^3\mathrm{d}x=\frac{1}{4}x{r}^3+\frac{3}{8}{a}^{2}xr+\frac{3}{8}{a}^4\ln (x+r)$

$\int r^5\mathrm{d}x=\frac{1}{6}x{r}^5+\frac{5}{24}{a}^{2}x{r}^3+\frac{5}{16}{a}^{4}xr+\frac{5}{16}{a}^6\ln (x+r)$

$\int xr\mathrm{d}x=\frac{1}{3}{r}^3$

$\int x{r}^3\mathrm{d}x=\frac{1}{5}{r}^5$

$\int x{r}^{2n+1}\mathrm{d}x=\frac{r^{2n+3}}{2n+3}$

$\int x^{2}r\mathrm{d}x=\frac{1}{4}x{r}^3-\frac{1}{8}{a}^{2}xr-\frac{1}{8}{a}^4\ln (x+r)$

$\int x^2{r}^3\mathrm{d}x=\frac{1}{6}x{r}^5-\frac{1}{24}{a}^{2}x{r}^3-\frac{1}{16}{a}^{4}xr-\frac{1}{16}{a}^6\ln (x+r)$

$\int x^{3}r\mathrm{d}x=\frac{1}{5}{r}^5-\frac{1}{3}{a}^2{r}^3$

$\int x^3{r}^3\mathrm{d}x=\frac{1}{7}{r}^7-\frac{1}{5}{a}^2{r}^5$

$\int x^3{r}^{2n+1}\mathrm{d}x=\frac{r^{2n+5}}{2n+5}-\frac{a^3{r}^{2n+3}}{2n+3}$

$\int x^{4}r\mathrm{d}x=\frac{1}{6}{x}^3{r}^3-\frac{1}{8}{a}^{2}x{r}^3+\frac{1}{16}{a}^{4}xr+\frac{1}{16}{a}^6\ln (x+r)$

$\int x^4{r}^3\mathrm{d}x=\frac{1}{8}{x}^3{r}^5-\frac{1}{16}{a}^{2}x{r}^5+\frac{1}{64}{a}^{4}x{r}^3+\frac{3}{128}{a}^{6}xr+\frac{3}{128}{a}^8\ln (x+r)$

$\int x^{5}r\mathrm{d}x=\frac{1}{7}{r}^7-\frac{2}{5}{a}^2{r}^5+\frac{1}{3}{a}^4{r}^3$

$\int x^5{r}^3\mathrm{d}x=\frac{1}{9}{r}^9-\frac{2}{7}{a}^2{r}^7+\frac{1}{5}{a}^4{r}^5$

$\int x^5{r}^{2n+1}\mathrm{d}x=\frac{r^{2n+7}}{2n+7}-\frac{2a^2{r}^{2n+5}}{2n+5}+\frac{a^4{r}^{2n+3}}{2n+3}$

$\int \frac{r\mathrm{d}x}{x}=r-a\ln \left|\frac{a+r}{x}\right|=r-a\mathrm{arsinh}\frac{a}{x}$

$\int \frac{r^3\mathrm{d}x}{x}=\frac{1}{3}{r}^3+a^{2}r-a^3\ln \left|\frac{a+r}{x}\right|$

$\int \frac{r^5\mathrm{d}x}{x}=\frac{1}{5}{r}^5+\frac{1}{3}{a}^2{r}^3+a^{4}r-a^5\ln \left|\frac{a+r}{x}\right|$

$\int \frac{r^7\mathrm{d}x}{x}=\frac{1}{7}{r}^7+\frac{1}{5}{a}^2{r}^5+\frac{1}{3}{a}^4{r}^3+a^{6}r-a^7\ln \left|\frac{a+r}{x}\right|$

$\int \frac{\mathrm{d}x}{r}=\mathrm{arsinh}\frac{x}{a}=\ln \left(\frac{x+r}{a}\right)$

$\int \frac{\mathrm{d}x}{r^3}=\frac{x}{a^{2}r}$

$\int \frac{x\mathrm{d}x}{r}=r$

$\int \frac{x\mathrm{d}x}{r^3}=-\frac{1}{r}$

$\int \frac{x^2\mathrm{d}x}{r}=\frac{1}{2}xr-\frac{1}{2}{a}^2\mathrm{arsinh}\frac{x}{a}=\frac{1}{2}xr-\frac{1}{2}{a}^2\ln \left(\frac{x+r}{a}\right)$

$\int \frac{\mathrm{d}x}{xr}=-\frac{1}{a}\mathrm{arsinh}\frac{a}{x}=-\frac{1}{a}\ln \left|\frac{a+r}{x}\right|$

Integralen met $s=\sqrt{x^2-a^2}$

Voor de volgende integralen is $x^2>a^2$.

$\int s\mathrm{d}x=\frac{1}{2}(xs-a^2\ln (x+s))$

$\int xs\mathrm{d}x=\frac{1}{3}{s}^3$

$\int \frac{s\mathrm{d}x}{x}=s-a\arccos \left|\frac{a}{x}\right|$

$\int \frac{\mathrm{d}x}{s}=\int \frac{\mathrm{d}x}{\sqrt{x^2-a^2}}=\ln \left|\frac{x+s}{a}\right|$

Houd hier rekening met het feit dat $\ln \left|\frac{x+s}{a}\right|=\mathrm{s}\mathrm{g}\mathrm{n}(x)\mathrm{arcosh}\left|\frac{x}{a}\right|=\frac{1}{2}\ln \left(\frac{x+s}{x-s}\right)$, waarbij alleen naar de positieve waarde moet worden gekeken, namelijk $\mathrm{arcosh}\left|\frac{x}{a}\right|$

$\int \frac{x\mathrm{d}x}{s}=s$

$\int \frac{x\mathrm{d}x}{s^3}=-\frac{1}{s}$

$\int \frac{x\mathrm{d}x}{s^5}=-\frac{1}{3s^3}$

$\int \frac{x\mathrm{d}x}{s^7}=-\frac{1}{5s^5}$

$\int \frac{x\mathrm{d}x}{s^{2n+1}}=-\frac{1}{(2n-1)s^{2n-1}}$

$\int \frac{x^{2m}\mathrm{d}x}{s^{2n+1}}=-\frac{1}{2n-1}\frac{x^{2m-1}}{s^{2n-1}}+\frac{2m-1}{2n-1}\int \frac{x^{2m-2}\mathrm{d}x}{s^{2n-1}}$

$\int \frac{x^2\mathrm{d}x}{s}=\frac{1}{2}xs+\frac{1}{2}{a}^2\ln \left|\frac{x+s}{a}\right|$

$\int \frac{x^2\mathrm{d}x}{s^3}=-\frac{x}{s}+\ln \left|\frac{x+s}{a}\right|$

$\int \frac{x^4\mathrm{d}x}{s}=\frac{1}{4}{x}^{3}s+\frac{3}{8}{a}^{2}xs+\frac{3}{8}{a}^4\ln \left|\frac{x+s}{a}\right|$

$\int \frac{x^4\mathrm{d}x}{s^3}=\frac{1}{2}xs-\frac{a^{2}x}{s}+\frac{3}{2}{a}^2\ln \left|\frac{x+s}{a}\right|$

$\int \frac{x^4\mathrm{d}x}{s^5}=-\frac{x}{s}-\frac{x^3}{3s^3}+\ln \left|\frac{x+s}{a}\right|$

$\int \frac{x^{2m}\mathrm{d}x}{s^{2n+1}}=(-1{)}^{n-m}\frac{1}{a^{2(n-m)}}\sum _{i=0}^{n-m-1}\frac{1}{2(m+i)+1}(\genfrac{}{}{0ex}{}{n-m-1}{i})\frac{x^{2(m+i)+1}}{s^{2(m+i)+1}}n>m\ge 0$

$\int \frac{\mathrm{d}x}{s^3}=-\frac{1}{a^2}\frac{x}{s}$

$\int \frac{\mathrm{d}x}{s^5}=\frac{1}{a^4}[\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3}]$

$\int \frac{\mathrm{d}x}{s^7}=-\frac{1}{a^6}[\frac{x}{s}-\frac{2}{3}\frac{x^3}{s^3}+\frac{1}{5}\frac{x^5}{s^5}]$

$\int \frac{\mathrm{d}x}{s^9}=\frac{1}{a^8}[\frac{x}{s}-\frac{3}{3}\frac{x^3}{s^3}+\frac{3}{5}\frac{x^5}{s^5}-\frac{1}{7}\frac{x^7}{s^7}]$

$\int \frac{x^2\mathrm{d}x}{s^5}=-\frac{1}{a^2}\frac{x^3}{3s^3}$

$\int \frac{x^2\mathrm{d}x}{s^7}=\frac{1}{a^4}[\frac{1}{3}\frac{x^3}{3s^3}-\frac{1}{5}\frac{x^5}{s^5}]$

$\int \frac{x^2\mathrm{d}x}{s^9}=-\frac{1}{a^6}[\frac{1}{3}\frac{x^3}{s^3}-\frac{2}{5}\frac{x^5}{s^5}+\frac{1}{7}\frac{x^7}{s^7}]$

Integralen waarbij $u=\sqrt{a^2-x^2}$

$\int u\mathrm{d}x=\frac{1}{2}(xu+a^2\arcsin \frac{x}{a})|x|\le |a|$

$\int xu\mathrm{d}x=-\frac{1}{3}{u}^3(|x|\le |a|)$

$\int x^{2}u\mathrm{d}x=-\frac{x}{4}{u}^3+\frac{a^2}{8}(xu+a^2\arcsin \frac{x}{a})|x|\le |a|$

$\int \frac{u\mathrm{d}x}{x}=u-a\ln \left|\frac{a+u}{x}\right||x|\le |a|$

$\int \frac{\mathrm{d}x}{u}=\arcsin \frac{x}{a}(|x|\le |a|)$

$\int \frac{x^2\mathrm{d}x}{u}=\frac{1}{2}(-xu+a^2\arcsin \frac{x}{a})|x|\le |a|$

$\int u\mathrm{d}x=\frac{1}{2}(xu-\mathrm{sgn}x\mathrm{arcosh}\left|\frac{x}{a}\right|)\text{voor }|x|\ge |a|$

$\int \frac{x}{u}\mathrm{d}x=-u|x|\le |a|$

Integralen waarbij $R=\sqrt{a{x}^2+bx+c}$

Neem aan dat $a{x}^2+bx+c$ niet kan worden geschreven als de verkorte vorm $(px+q{)}^2$

$\int \frac{\mathrm{d}x}{R}=\frac{1}{\sqrt{a}}\ln |2\sqrt{a}R+2ax+b|\text{voor }a>0$

$\int \frac{\mathrm{d}x}{R}=\frac{1}{\sqrt{a}}\mathrm{arsinh}\frac{2ax+b}{\sqrt{4ac-b^2}}\text{voor }a>0,4ac-b^2>0$

$\int \frac{\mathrm{d}x}{R}=\frac{1}{\sqrt{a}}\ln |2ax+b|\text{voor }a>0,4ac-b^2=0$

$\int \frac{\mathrm{d}x}{R}=-\frac{1}{\sqrt{-a}}\arcsin \frac{2ax+b}{\sqrt{b^2-4ac}}\text{voor }a<0,4ac-b^2<0,|2ax+b|<\sqrt{b^2-4ac}$

$\int \frac{\mathrm{d}x}{R^3}=\frac{4ax+2b}{(4ac-b^2)R}$

$\int \frac{\mathrm{d}x}{R^5}=\frac{4ax+2b}{3(4ac-b^2)R}(\frac{1}{R^2}+\frac{8a}{4ac-b^2})$

$\int \frac{\mathrm{d}x}{R^{2n+1}}=\frac{2}{(2n-1)(4ac-b^2)}(\frac{2ax+b}{R^{2n-1}}+4a(n-1)\int \frac{\mathrm{d}x}{R^{2n-1}})$

$\int \frac{x}{R}\mathrm{d}x=\frac{R}{a}-\frac{b}{2a}\int \frac{\mathrm{d}x}{R}$

$\int \frac{x}{R^3}\mathrm{d}x=-\frac{2bx+4c}{(4ac-b^2)R}$

$\int \frac{x}{R^{2n+1}}\mathrm{d}x=-\frac{1}{(2n-1)a{R}^{2n-1}}-\frac{b}{2a}\int \frac{\mathrm{d}x}{R^{2n+1}}$

$\int \frac{\mathrm{d}x}{xR}=-\frac{1}{\sqrt{c}}\ln \left(\frac{2\sqrt{c}R+bx+2c}{x}\right)$

$\int \frac{\mathrm{d}x}{xR}=-\frac{1}{\sqrt{c}}\mathrm{arsinh}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right)$

Integralen waarbij $S=\sqrt{ax+b}$

$\int S\mathrm{d}x=\frac{2S^3}{3a}$

$\int \frac{\mathrm{d}x}{S}=\frac{2S}{a}$

$\int \frac{\mathrm{d}x}{xS}=\{\begin{array}{c}\text{voor }b>0,ax>0\\ -\frac{2}{\sqrt{b}}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{S}{\sqrt{b}}\right)& \text{voor }b>0,ax<0\\ \frac{2}{\sqrt{-b}}\arctan \left(\frac{S}{\sqrt{-b}}\right)& \text{voor }b<0\end{array}$

$\int \frac{S}{x}\mathrm{d}x=\{\begin{array}{cc}2(S-\sqrt{b}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\frac{S}{\sqrt{b}}\right))& \text{voor }b>0,ax>0\\ 2(S-\sqrt{b}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{S}{\sqrt{b}}\right))& \text{voor }b>0,ax<0\\ 2(S-\sqrt{-b}\arctan \left(\frac{S}{\sqrt{-b}}\right))& \text{voor }b<0\end{array}$

$\int \frac{x^n}{S}\mathrm{d}x=\frac{2}{a(2n+1)}(x^{n}S-bn\int \frac{x^{n-1}}{S}\mathrm{d}x)$

$\int x^{n}S\mathrm{d}x=\frac{2}{a(2n+3)}(x^n{S}^3-nb\int x^{n-1}S\mathrm{d}x)$

$\int \frac{1}{x^{n}S}\mathrm{d}x=-\frac{1}{b(n-1)}(\frac{S}{x^{n-1}}+(n-\frac{3}{2})a\int \frac{\mathrm{d}x}{x^{n-1}S})$

Literatuur

• Sjabloon:Aut, Sjabloon:Aut, Sjabloon:Aut en Sjabloon:Aut. Table of Integrals, Series, and Products, 2007. met te downloaden pdf's, ISBN 978-0-12-373637-6.