Verskil tussen weergawes van "Lys van integrale"

Opruim
k (Verplasing van 1 interwikiskakels wat op Wikidata beskikbaar is op d:q423189)
(Opruim)
Vir die doeleindes van hierdie lys word ''K'' as arbitrêre-integrasiekonstante gebruik.
 
== Reëls by die integreer van algemene funksies ==
: <math>\int af(x)\,dx = a\int f(x)\,dx \qquad\mbox{(}a \mbox{ konstant)}\,\!</math>
: <math>\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,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>
: <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)\over f(x)}\,dx= \ln{\left|f(x)\right|} + K </math>
: <math>\int {f'(x) f(x)}\,dx= {1 \over 2} [ f(x) ]^2 + K </math>
 
== Integrale van eenvoudige funksies ==
 
=== Rasionale funksies ===
: <math>\int \,{\rm d}x = x + K</math>
: <math>\int x^n\,{\rm d}x = \frac{x^{n+1}}{n+1} + K\qquad\mbox{ mits }n \ne -1</math>
: <math>\int {dx \over x} = \ln{\left|x\right|} + K</math>
: <math>\int {dx \over {a^2+x^2}} = {1 \over a}\mbox{(bgtan)} {x \over a} + K</math>
 
=== Irrasionale funksies ===
: <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}} = \cos^{-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>
 
=== Logaritmes ===
: <math>\int \ln {x}\,dx = x \ln {x} - x + K</math>
: <math>\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + K</math>
 
=== Eksponensiaalfunksies ===
: <math>\int e^x\,dx = e^x + K</math>
: <math>\int a^x\,dx = \frac{a^x}{\ln{a}} + K</math>
 
=== Trigonometriese funksies ===
: <math>\int \sin{x}\, dx = -\cos{x} + K</math>
: <math>\int \cos{x}\, dx = \sin{x} + K</math>
: <math>\int \tan{x} \, dx = \ln{\left| \sec {x} \right|} + K</math>
: <math>\int \cot{x} \, dx = -\ln{\left| \csc{x} \right|} + K</math>
: <math>\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + K</math>
: <math>\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + K</math>
: <math>\int \sec^2 x \, dx = \tan x +K</math>
: <math>\int \csc^2 x \, dx = -\cot x + K</math>
: <math>\int \sec{x} \, \tan{x} \, dx = \sec{x} + K</math>
: <math>\int \csc{x} \, \cot{x} \, dx = - \csc{x} + K</math>
: <math>\int \sin^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>
: <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 \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 \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 \mbox{bgtan}{x} \, dx = x \, \mbox{bgtan}{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + K</math>
 
=== Hiperboliese funksies ===
: <math>\int \sinh x \, dx = \cosh x + K</math>
: <math>\int \cosh x \, dx = \sinh x + K</math>
: <math>\int \tanh x \, dx = \ln| \cosh x | + K</math>
: <math>\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + K</math>
: <math>\int \mbox{sech}\,x \, dx = \mbox{bgtan}(\sinh x) + K</math>
: <math>\int \coth x \, dx = \ln| \sinh x | + K</math>
: <math>\int \mbox{sech}^2 x\, dx = \tanh x + K</math>
 
=== Inverse hiperboliese funksies ===
: <math>\int \sinh ^{-1} x \, dx = x \sinh ^{-1} x - \sqrt{x^2+1} + K</math>
: <math>\int \cosh ^{-1} x \, dx = x \cosh ^{-1} x - \sqrt{x^2-1} + K</math>
: <math>\int \tanh ^{-1} x \, dx = x \tanh ^{-1} x + \frac{1}{2}\log{(1-x^2)} + 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>
: <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 \coth ^{-1} x \, dx = x \coth ^{-1} x+ \frac{1}{2}\log{(x^2-1)} + K</math>
 
=== Bepaalde integrale sonder geslote-vorm afgeleides ===
: <math>\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi</math>
 
: <math>\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi</math> (die [[Gaussiese integraal]])
 
: <math>\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}</math>
 
: <math>\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}</math>
 
: <math>\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{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>)
 
: <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^\infty\frac{\sin^2{x}}{x^2}\,dx=\frac{\pi}{2}</math>
 
: <math>\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)</math> (waar <math>\Gamma(z)</math> die [[Gamma funksie]] 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.)
 
: <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 + y \sin \theta} d \theta = 2 \pi I_{0} \sqrt{x^2 + y^2} </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>.
 
: <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>
 
== 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 [http[://en.wikipedia.org/wiki/Table_of_integrals:Table of integrals|Table of integrals]]
 
[[Kategorie:Lyste|Integrale]]