Lys van integrale: Verskil tussen weergawes
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Vir die doeleindes van hierdie lys word ''K'' as arbitrêre-integrasiekonstante gebruik. |
Vir die doeleindes van hierdie lys word ''K'' as arbitrêre-integrasiekonstante gebruik. |
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==Reëls by die integreer van algemene funksies== |
== Reëls by die integreer van algemene funksies == |
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:<math>\int af(x)\,dx = a\int f(x)\,dx \qquad\mbox{(}a \mbox{ konstant)}\,\!</math> |
: <math>\int af(x)\,dx = a\int f(x)\,dx \qquad\mbox{(}a \mbox{ konstant)}\,\!</math> |
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:<math>\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx</math> |
: <math>\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx</math> |
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:<math>\int f(x)g(x)\,dx = f(x)\int g(x)\,dx - \int \left[f'(x) \left(\int g(x)\,dx\right)\right]\,dx</math> |
: <math>\int f(x)g(x)\,dx = f(x)\int g(x)\,dx - \int \left[f'(x) \left(\int g(x)\,dx\right)\right]\,dx</math> |
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:<math>\int [f(x)]^n f'(x)\,dx = {[f(x)]^{n+1} \over n+1} + K \qquad\mbox{(vir } n\neq -1\mbox{)}\,\! </math> |
: <math>\int [f(x)]^n f'(x)\,dx = {[f(x)]^{n+1} \over n+1} + K \qquad\mbox{(vir } n\neq -1\mbox{)}\,\! </math> |
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:<math>\int {f'(x)\over f(x)}\,dx= \ln{\left|f(x)\right|} + K </math> |
: <math>\int {f'(x)\over f(x)}\,dx= \ln{\left|f(x)\right|} + K </math> |
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:<math>\int {f'(x) f(x)}\,dx= {1 \over 2} [ f(x) ]^2 + K </math> |
: <math>\int {f'(x) f(x)}\,dx= {1 \over 2} [ f(x) ]^2 + K </math> |
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==Integrale van eenvoudige funksies== |
== Integrale van eenvoudige funksies == |
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===Rasionale funksies=== |
=== Rasionale funksies === |
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:<math>\int \,{\rm d}x = x + K</math> |
: <math>\int \,{\rm d}x = x + K</math> |
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:<math>\int x^n\,{\rm d}x = \frac{x^{n+1}}{n+1} + K\qquad\mbox{ mits }n \ne -1</math> |
: <math>\int x^n\,{\rm d}x = \frac{x^{n+1}}{n+1} + K\qquad\mbox{ mits }n \ne -1</math> |
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:<math>\int {dx \over x} = \ln{\left|x\right|} + K</math> |
: <math>\int {dx \over x} = \ln{\left|x\right|} + K</math> |
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:<math>\int {dx \over {a^2+x^2}} = {1 \over a}\mbox{(bgtan)} {x \over a} + K</math> |
: <math>\int {dx \over {a^2+x^2}} = {1 \over a}\mbox{(bgtan)} {x \over a} + K</math> |
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===Irrasionale funksies=== |
=== Irrasionale funksies === |
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:<math>\int {dx \over \sqrt{a^2-x^2}} = \sin^{-1} {x \over a} + K</math> |
: <math>\int {dx \over \sqrt{a^2-x^2}} = \sin^{-1} {x \over a} + K</math> |
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:<math>\int {-dx \over \sqrt{a^2-x^2}} = \cos^{-1} {x \over a} + K</math> |
: <math>\int {-dx \over \sqrt{a^2-x^2}} = \cos^{-1} {x \over a} + K</math> |
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:<math>\int {dx \over x \sqrt{x^2-a^2}} = {1 \over a} \sec^{-1} {|x| \over a} + K</math> |
: <math>\int {dx \over x \sqrt{x^2-a^2}} = {1 \over a} \sec^{-1} {|x| \over a} + K</math> |
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===Logaritmes=== |
=== Logaritmes === |
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:<math>\int \ln {x}\,dx = x \ln {x} - x + K</math> |
: <math>\int \ln {x}\,dx = x \ln {x} - x + K</math> |
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:<math>\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + K</math> |
: <math>\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + K</math> |
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===Eksponensiaalfunksies=== |
=== Eksponensiaalfunksies === |
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:<math>\int e^x\,dx = e^x + K</math> |
: <math>\int e^x\,dx = e^x + K</math> |
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:<math>\int a^x\,dx = \frac{a^x}{\ln{a}} + K</math> |
: <math>\int a^x\,dx = \frac{a^x}{\ln{a}} + K</math> |
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===Trigonometriese funksies=== |
=== Trigonometriese funksies === |
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:<math>\int \sin{x}\, dx = -\cos{x} + K</math> |
: <math>\int \sin{x}\, dx = -\cos{x} + K</math> |
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:<math>\int \cos{x}\, dx = \sin{x} + K</math> |
: <math>\int \cos{x}\, dx = \sin{x} + K</math> |
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:<math>\int \tan{x} \, dx = \ln{\left| \sec {x} \right|} + K</math> |
: <math>\int \tan{x} \, dx = \ln{\left| \sec {x} \right|} + K</math> |
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:<math>\int \cot{x} \, dx = -\ln{\left| \csc{x} \right|} + K</math> |
: <math>\int \cot{x} \, dx = -\ln{\left| \csc{x} \right|} + K</math> |
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:<math>\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + K</math> |
: <math>\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + K</math> |
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:<math>\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + K</math> |
: <math>\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + K</math> |
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:<math>\int \sec^2 x \, dx = \tan x +K</math> |
: <math>\int \sec^2 x \, dx = \tan x +K</math> |
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:<math>\int \csc^2 x \, dx = -\cot x + K</math> |
: <math>\int \csc^2 x \, dx = -\cot x + K</math> |
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:<math>\int \sec{x} \, \tan{x} \, dx = \sec{x} + K</math> |
: <math>\int \sec{x} \, \tan{x} \, dx = \sec{x} + K</math> |
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:<math>\int \csc{x} \, \cot{x} \, dx = - \csc{x} + K</math> |
: <math>\int \csc{x} \, \cot{x} \, dx = - \csc{x} + K</math> |
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:<math>\int \sin^2 x \, dx = \frac{1}{2}(x - \sin x \cos x) + K</math> |
: <math>\int \sin^2 x \, dx = \frac{1}{2}(x - \sin x \cos x) + K</math> |
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:<math>\int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + K</math> |
: <math>\int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + K</math> |
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:<math>\int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + K</math> |
: <math>\int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + K</math> |
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:<math>\int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx</math> |
: <math>\int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx</math> |
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:<math>\int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx</math> |
: <math>\int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx</math> |
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:<math>\int \mbox{bgtan}{x} \, dx = x \, \mbox{bgtan}{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + K</math> |
: <math>\int \mbox{bgtan}{x} \, dx = x \, \mbox{bgtan}{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + K</math> |
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===Hiperboliese funksies=== |
=== Hiperboliese funksies === |
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:<math>\int \sinh x \, dx = \cosh x + K</math> |
: <math>\int \sinh x \, dx = \cosh x + K</math> |
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:<math>\int \cosh x \, dx = \sinh x + K</math> |
: <math>\int \cosh x \, dx = \sinh x + K</math> |
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:<math>\int \tanh x \, dx = \ln| \cosh x | + K</math> |
: <math>\int \tanh x \, dx = \ln| \cosh x | + K</math> |
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:<math>\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + K</math> |
: <math>\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + K</math> |
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:<math>\int \mbox{sech}\,x \, dx = \mbox{bgtan}(\sinh x) + K</math> |
: <math>\int \mbox{sech}\,x \, dx = \mbox{bgtan}(\sinh x) + K</math> |
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:<math>\int \coth x \, dx = \ln| \sinh x | + K</math> |
: <math>\int \coth x \, dx = \ln| \sinh x | + K</math> |
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:<math>\int \mbox{sech}^2 x\, dx = \tanh x + K</math> |
: <math>\int \mbox{sech}^2 x\, dx = \tanh x + K</math> |
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===Inverse hiperboliese funksies=== |
=== Inverse hiperboliese funksies === |
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: <math>\int \sinh ^{-1} x \, dx = x \sinh ^{-1} x - \sqrt{x^2+1} + K</math> |
: <math>\int \sinh ^{-1} x \, dx = x \sinh ^{-1} x - \sqrt{x^2+1} + K</math> |
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: <math>\int \cosh ^{-1} x \, dx = x \cosh ^{-1} x - \sqrt{x^2-1} + K</math> |
: <math>\int \cosh ^{-1} x \, dx = x \cosh ^{-1} x - \sqrt{x^2-1} + K</math> |
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: <math>\int \tanh ^{-1} x \, dx = x \tanh ^{-1} x + \frac{1}{2}\log{(1-x^2)} + K</math> |
: <math>\int \tanh ^{-1} x \, dx = x \tanh ^{-1} x + \frac{1}{2}\log{(1-x^2)} + K</math> |
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: <math>\int \mbox{csch} ^{-1} \,x \, dx = x \mbox{csch} ^{-1} x + \log{\left[x\left(\sqrt{1+\frac{1}{x^2}} + 1\right)\right]} + K</math> |
: <math>\int \mbox{csch} ^{-1} \,x \, dx = x \mbox{csch} ^{-1} x + \log{\left[x\left(\sqrt{1+\frac{1}{x^2}} + 1\right)\right]} + K</math> |
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: <math>\int \mbox{sech} ^{-1} \,x \, dx = x \mbox{sech} ^{-1} x - \mbox{bgtan}{\left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)} + K</math> |
: <math>\int \mbox{sech} ^{-1} \,x \, dx = x \mbox{sech} ^{-1} x - \mbox{bgtan}{\left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)} + K</math> |
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: <math>\int \coth ^{-1} x \, dx = x \coth ^{-1} x+ \frac{1}{2}\log{(x^2-1)} + K</math> |
: <math>\int \coth ^{-1} x \, dx = x \coth ^{-1} x+ \frac{1}{2}\log{(x^2-1)} + K</math> |
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===Bepaalde integrale sonder geslote-vorm afgeleides=== |
=== Bepaalde integrale sonder geslote-vorm afgeleides === |
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:<math>\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi</math> |
: <math>\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi</math> |
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:<math>\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi</math> (die [[Gaussiese integraal]]) |
: <math>\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi</math> (die [[Gaussiese integraal]]) |
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:<math>\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}</math> |
: <math>\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}</math> |
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:<math>\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}</math> |
: <math>\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}</math> |
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:<math>\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}</math> |
: <math>\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}</math> |
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:<math>\int_0^\frac{\pi}{2}\sin^n{x}\,dx=\int_0^\frac{\pi}{2}\cos^n{x}\,dx=\frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot (n-1)}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot n}\frac{\pi}{2}</math> (mits ''n'' 'n ewe heelgetal en <math> \scriptstyle{n \ge 2}</math>) |
: <math>\int_0^\frac{\pi}{2}\sin^n{x}\,dx=\int_0^\frac{\pi}{2}\cos^n{x}\,dx=\frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot (n-1)}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot n}\frac{\pi}{2}</math> (mits ''n'' 'n ewe heelgetal en <math> \scriptstyle{n \ge 2}</math>) |
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:<math>\int_0^\frac{\pi}{2}\sin^n{x}\,dx=\int_0^\frac{\pi}{2}\cos^n{x}\,dx=\frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot (n-1)}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot n}</math> (mits ''n'' 'n onewe heelgetal en <math> \scriptstyle{n \ge 3} </math>) |
: <math>\int_0^\frac{\pi}{2}\sin^n{x}\,dx=\int_0^\frac{\pi}{2}\cos^n{x}\,dx=\frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot (n-1)}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot n}</math> (mits ''n'' 'n onewe heelgetal en <math> \scriptstyle{n \ge 3} </math>) |
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:<math>\int_0^\infty\frac{\sin^2{x}}{x^2}\,dx=\frac{\pi}{2}</math> |
: <math>\int_0^\infty\frac{\sin^2{x}}{x^2}\,dx=\frac{\pi}{2}</math> |
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:<math>\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)</math> (waar <math>\Gamma(z)</math> die [[Gamma funksie]] is) |
: <math>\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)</math> (waar <math>\Gamma(z)</math> die [[Gamma funksie]] is) |
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:<math>\int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx=\sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right]</math> (waar <math>\exp[u]</math> die [[eksponensiaalfunksie]] <math>e^u</math> is.) |
: <math>\int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx=\sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right]</math> (waar <math>\exp[u]</math> die [[eksponensiaalfunksie]] <math>e^u</math> is.) |
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:<math>\int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x)</math> (waar <math>I_{0}(x)</math> die gewysigde [[Bessel funksie]] van die eerste tipe is.) |
: <math>\int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x)</math> (waar <math>I_{0}(x)</math> die gewysigde [[Bessel funksie]] van die eerste tipe is.) |
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:<math>\int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \sqrt{x^2 + y^2} </math> |
: <math>\int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \sqrt{x^2 + y^2} </math> |
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:<math>\int_{-\infty}^{\infty}{(1 + x^2/\nu)^{-(\nu + 1)/2}dx} = \frac { \sqrt{\nu \pi} \ \Gamma(\nu/2)} {\Gamma((\nu + 1)/2))}\,</math> (<math>\nu > 0\,</math>. |
: <math>\int_{-\infty}^{\infty}{(1 + x^2/\nu)^{-(\nu + 1)/2}dx} = \frac { \sqrt{\nu \pi} \ \Gamma(\nu/2)} {\Gamma((\nu + 1)/2))}\,</math> (<math>\nu > 0\,</math>. |
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:<math>\int_a^b{f(x)\,dx} = (b - a) \sum\limits_{n = 1}^\infty {\sum\limits_{m = 1}^{2^n - 1} {\left( { - 1} \right)^{m + 1} } } 2^{ - n} f(a + m\left( {b - a} \right)2^{-n} )</math> |
: <math>\int_a^b{f(x)\,dx} = (b - a) \sum\limits_{n = 1}^\infty {\sum\limits_{m = 1}^{2^n - 1} {\left( { - 1} \right)^{m + 1} } } 2^{ - n} f(a + m\left( {b - a} \right)2^{-n} )</math> |
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==Verwysings== |
== Verwysings == |
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#Stewart, J. (2003). ''Single Variable Calculus''. (5th ed.). Belmont, USA: Thomson Learning. |
# Stewart, J. (2003). ''Single Variable Calculus''. (5th ed.). Belmont, USA: Thomson Learning. |
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#Groenewald, G.J., Hitge, M. (2005). ''Analise II Studiegids vir WISK121A''. Potchefstroom: Noordwes-Universiteit. |
# Groenewald, G.J., Hitge, M. (2005). ''Analise II Studiegids vir WISK121A''. Potchefstroom: Noordwes-Universiteit. |
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#Jordan, D.W., Smith, P. (2002). ''Mathematical techniques: An introduction for the engineering, physical and mathematical sciences''. USA: Oxford University Press. |
# Jordan, D.W., Smith, P. (2002). ''Mathematical techniques: An introduction for the engineering, physical and mathematical sciences''. USA: Oxford University Press. |
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==Aantekeninge== |
== Aantekeninge == |
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*''Sien ook [[Lys van afgeleides]] |
* ''Sien ook [[Lys van afgeleides]] |
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*''Hierdie artikel is 'n vertaling van die Engelse Wikipedia artikel [ |
* ''Hierdie artikel is 'n vertaling van die Engelse Wikipedia artikel [[:en:Table of integrals|Table of integrals]] |
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[[Kategorie:Lyste|Integrale]] |
[[Kategorie:Lyste|Integrale]] |
Wysiging soos op 07:41, 29 November 2015
Integrasie is een van die hoofbewerkings van calculus. Vir differensiasie kan die eenvoudiger dele van 'n funksie maklik gedifferensieer word, wat differensiasie dan vergemaklik, maar dit kan egter nie met integrasie gedoen word nie. Vir gevalle waar daar met komplekse funksies gewerk word, is dit makliker om 'n lys van integrale byderhand te hou.
Vir die doeleindes van hierdie lys word K as arbitrêre-integrasiekonstante gebruik.
Reëls by die integreer van algemene funksies
Integrale van eenvoudige funksies
Rasionale funksies
Irrasionale funksies
Logaritmes
Eksponensiaalfunksies
Trigonometriese funksies
Hiperboliese funksies
Inverse hiperboliese funksies
Bepaalde integrale sonder geslote-vorm afgeleides
- (die Gaussiese integraal)
- (mits n 'n ewe heelgetal en )
- (mits n 'n onewe heelgetal en )
- (waar die Gamma funksie is)
- (waar die eksponensiaalfunksie is.)
- (waar die gewysigde Bessel funksie van die eerste tipe is.)
- (.
Verwysings
- Stewart, J. (2003). Single Variable Calculus. (5th ed.). Belmont, USA: Thomson Learning.
- Groenewald, G.J., Hitge, M. (2005). Analise II Studiegids vir WISK121A. Potchefstroom: Noordwes-Universiteit.
- Jordan, D.W., Smith, P. (2002). Mathematical techniques: An introduction for the engineering, physical and mathematical sciences. USA: Oxford University Press.
Aantekeninge
- Sien ook Lys van afgeleides
- Hierdie artikel is 'n vertaling van die Engelse Wikipedia artikel Table of integrals