List of integrals of irrational functions and User:84user/Images uploaded: Difference between pages

Add links
From Wikipedia, the free encyclopedia
(Difference between pages)
Content deleted Content added
VisualWikitext
 
m cats
 
Line 1: Line 1:
[http://en.wikipedia.org/w/index.php?limit=50&title=Special%3AContributions&contribs=user&target=84user&namespace=6&year=&month=-1 My edits and uploads to the Image namespace]
__NOTOC__
The following is a list of [[integral]]s ([[antiderivative]] functions) of [[irrational function]]s. For a complete list of integral functions, see [[lists of integrals]].


<gallery caption="Images I created" widths="100px" heights="80px" perrow="4">
== Integrals involving <math>r = \sqrt{x^2+a^2}</math> ==
Image:Credit default swaps by quality size coloured sp percent years.png|Composition of the United States 15.5 trillion US dollar CDS market at the end of 2008 Q2. Green tints indicate Prime asset CDSs, and red tints, reddish tints indicate sub-prime asset CDSs. Numbers followed by "Y" indicate the years until maturity.
Image:Wikitable overlapping HTML table.png|Table render bug in Firefox 1 and 2
Image:Mozilla bug with commonscat in Category.png|Overlapping bug of Mozilla browsers up to but not including Firefox 3.0.1
Image:Mozilla bug with commonscat in Category crop.png|Cropped image of Mozilla browser overlapping bug
Image:Arles portrait bust right then left disrupts flow narrow window disrupts.png‎|image affecting text flow
Image:Arles portrait bust all right narrow is clean.png‎|images at right look better
Image:Arles portrait bust right then left disrupts flow.png|images disrupt flow
Image:Arles portrait bust all right.png|images at right wide
</gallery>


<gallery caption="Images I found" widths="100px" heights="80px" perrow="4">
: <math>\int r \;dx = \frac{1}{2}\left(x r +a^2\,\ln\left(x+r\right)\right)</math><!-- (1.1) [Abramowitz & Stegun p13 3.3.41] + verified by differentiation -->
Image:F.6 Forlanini Airship.jpg‎|F.6 dirigible built by Enrico Forlanini in 1918
</gallery>


[[Category:Wikipedia image galleries|mixed]]
: <math>\int r^3 \;dx = \frac{1}{4}xr^3+\frac{1}{8}3a^2xr+\frac{3}{8}a^4\ln\left(x+r\right)</math>
[[Category:User page galleries|84user]]

: <math>\int r^5 \; dx = \frac{1}{6}xr^5+\frac{5}{24}a^2xr^3+\frac{5}{16}a^4xr+\frac{5}{16}a^6\ln\left(x+r\right)</math>

: <math>\int x r\;dx=\frac{r^3}{3}</math>

: <math>\int x r^3\;dx=\frac{r^5}{5}</math>

: <math>\int x r^{2n+1}\;dx=\frac{r^{2n+3}}{2n+3} </math>

: <math>\int x^2 r\;dx= \frac{xr^3}{4}-\frac{a^2xr}{8}-\frac{a^4}{8}\ln\left(x+r\right)</math>

: <math>\int x^2 r^3\;dx= \frac{xr^5}{6}-\frac{a^2xr^3}{24}-\frac{a^4xr}{16}-\frac{a^6}{16}\ln\left(x+r\right)</math>

: <math>\int x^3 r \; dx = \frac{r^5}{5} - \frac{a^2 r^3}{3}</math>

: <math>\int x^3 r^3 \; dx = \frac{r^7}{7}-\frac{a^2r^5}{5} </math>

: <math>\int x^3 r^{2n+1} \; dx = \frac{r^{2n+5}}{2n+5} - \frac{a^3 r^{2n+3}}{2n+3}</math>

: <math>\int x^4 r\;dx= \frac{x^3r^3}{6}-\frac{a^2xr^3}{8}+\frac{a^4xr}{16}+\frac{a^6}{16}\ln\left(x+r\right)</math>

: <math>\int x^4 r^3\;dx= \frac{x^3r^5}{8}-\frac{a^2xr^5}{16}+\frac{a^4xr^3}{64}+\frac{3a^6xr}{128}+\frac{3a^8}{128}\ln\left(x+r\right)</math>

: <math>\int x^5 r \; dx = \frac{r^7}{7} - \frac{2 a^2 r^5}{5} + \frac{a^4 r^3}{3}</math>

: <math>\int x^5 r^3 \; dx = \frac{r^9}{9} - \frac{2 a^2 r^7}{7} + \frac{a^4 r^5}{5}</math>

: <math>\int x^5 r^{2n+1} \; dx = \frac{r^{2n+7}}{2n+7} - \frac{2a^2r^{2n+5}}{2n+5}+\frac{a^4 r^{2n+3}}{2n+3} </math>

: <math>\int\frac{r\;dx}{x} = r-a\ln\left|\frac{a+r}{x}\right| = r - a\, \operatorname{arsinh}\frac{a}{x}</math>

: <math>\int\frac{r^3\;dx}{x} = \frac{r^3}{3}+a^2r-a^3\ln\left|\frac{a+r}{x}\right|</math>

: <math>\int\frac{r^5\;dx}{x} = \frac{r^5}{5}+\frac{a^2r^3}{3}+a^4r-a^5\ln\left|\frac{a+r}{x}\right|</math>

: <math>\int\frac{r^7\;dx}{x} = \frac{r^7}{7}+\frac{a^2r^5}{5}+\frac{a^4r^3}{3}+a^6r-a^7\ln\left|\frac{a+r}{x}\right|</math>

: <math>\int\frac{dx}{r} = \operatorname{arsinh}\frac{x}{a} = \ln\left|x+r\right|</math>

: <math>\int\frac{dx}{r^3} = \frac{x}{a^2r}</math>

: <math>\int\frac{x\,dx}{r} = r</math>

: <math>\int\frac{x\,dx}{r^3} = -\frac{1}{r}</math>

: <math>\int\frac{x^2\;dx}{r} = \frac{x}{2}r-\frac{a^2}{2}\,\operatorname{arsinh}\frac{x}{a} = \frac{x}{2}r-\frac{a^2}{2}\ln\left|x+r\right|</math>

: <math>\int\frac{dx}{xr} = -\frac{1}{a}\,\operatorname{arsinh}\frac{a}{x} = -\frac{1}{a}\ln\left|\frac{a+r}{x}\right|</math>

== Integrals involving <math>s = \sqrt{x^2-a^2}</math>==
Assume <math>(x^2>a^2)</math>, for <math>(x^2<a^2)</math>, see next section:
: <math>\int xs\;dx = \frac{1}{3}s^3</math>

: <math>\int\frac{s\;dx}{x} = s - a\arccos\left|\frac{a}{x}\right|</math>

: <math>\int\frac{dx}{s} = \int\frac{dx}{\sqrt{x^2-a^2}} =\ln\left|\frac{x+s}{a}\right|</math>
Note that <math>\ln\left|\frac{x+s}{a}\right|
=\mathrm{sgn}(x)\,\operatorname{arcosh}\left|\frac{x}{a}\right|
=\frac{1}{2}\ln\left(\frac{x+s}{x-s}\right)</math>, where the positive value of <math>\operatorname{arcosh}\left|\frac{x}{a}\right|</math> is to be taken.

: <math>\int\frac{x\;dx}{s} = s</math>

: <math>\int\frac{x\;dx}{s^3} = -\frac{1}{s}</math>

: <math>\int\frac{x\;dx}{s^5} = -\frac{1}{3s^3}</math>

: <math>\int\frac{x\;dx}{s^7} = -\frac{1}{5s^5}</math>

: <math>\int\frac{x\;dx}{s^{2n+1}} = -\frac{1}{(2n-1)s^{2n-1}} </math>

: <math>\int\frac{x^{2m}\;dx}{s^{2n+1}}
= -\frac{1}{2n-1}\frac{x^{2m-1}}{s^{2n-1}}+\frac{2m-1}{2n-1}\int\frac{x^{2m-2}\;dx}{s^{2n-1}}
</math>

: <math>\int\frac{x^2\;dx}{s}
= \frac{xs}{2}+\frac{a^2}{2}\ln\left|\frac{x+s}{a}\right|</math>

: <math>\int\frac{x^2\;dx}{s^3}
= -\frac{x}{s}+\ln\left|\frac{x+s}{a}\right|</math>

: <math>\int\frac{x^4\;dx}{s}
= \frac{x^3s}{4}+\frac{3}{8}a^2xs+\frac{3}{8}a^4\ln\left|\frac{x+s}{a}\right| </math>

: <math>\int\frac{x^4\;dx}{s^3}
= \frac{xs}{2}-\frac{a^2x}{s}+\frac{3}{2}a^2\ln\left|\frac{x+s}{a}\right| </math>

: <math>\int\frac{x^4\;dx}{s^5}
= -\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3}+\ln\left|\frac{x+s}{a}\right| </math>

: <math>\int\frac{x^{2m}\;dx}{s^{2n+1}}
= (-1)^{n-m}\frac{1}{a^{2(n-m)}}\sum_{i=0}^{n-m-1}\frac{1}{2(m+i)+1}{n-m-1 \choose i}\frac{x^{2(m+i)+1}}{s^{2(m+i)+1}}\qquad\mbox{(}n>m\ge0\mbox{)}</math>

: <math>\int\frac{dx}{s^3}=-\frac{1}{a^2}\frac{x}{s}</math>

: <math>\int\frac{dx}{s^5}=\frac{1}{a^4}\left[\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3}\right]</math>

: <math>\int\frac{dx}{s^7}
=-\frac{1}{a^6}\left[\frac{x}{s}-\frac{2}{3}\frac{x^3}{s^3}+\frac{1}{5}\frac{x^5}{s^5}\right]</math>

: <math>\int\frac{dx}{s^9}
=\frac{1}{a^8}\left[\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}\right]</math>

: <math>\int\frac{x^2\;dx}{s^5}=-\frac{1}{a^2}\frac{x^3}{3s^3}</math>

: <math>\int\frac{x^2\;dx}{s^7}
= \frac{1}{a^4}\left[\frac{1}{3}\frac{x^3}{s^3}-\frac{1}{5}\frac{x^5}{s^5}\right]</math>

: <math>\int\frac{x^2\;dx}{s^9}
= -\frac{1}{a^6}\left[\frac{1}{3}\frac{x^3}{s^3}-\frac{2}{5}\frac{x^5}{s^5}+\frac{1}{7}\frac{x^7}{s^7}\right]</math>

== Integrals involving <math>u = \sqrt{a^2-x^2}</math>==
: <math>\int u \;dx = \frac{1}{2}\left(xu+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}</math><!-- (3.1) [Abramowitz & Stegun p13 3.3.45] -->

: <math>\int xu\;dx = -\frac{1}{3} u^3 \qquad\mbox{(}|x|\leq|a|\mbox{)}</math>

: <math>\int\frac{u\;dx}{x} = u-a\ln\left|\frac{a+u}{x}\right| \qquad\mbox{(}|x|\leq|a|\mbox{)}</math>

: <math>\int\frac{dx}{u} = \arcsin\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)}</math><!-- (3.4) [Abramowitz & Stegun p13 3.3.44] -->

: <math>\int\frac{x^2\;dx}{u} = \frac{1}{2}\left(-xu+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}</math><!-- (3.5) [need reference] - verified by differentiation only -->

: <math>\int u\;dx = \frac{1}{2}\left(xu-\sgn x\,\operatorname{arcosh}\left|\frac{x}{a}\right|\right) \qquad\mbox{(for }|x|\ge|a|\mbox{)}</math>

== Integrals involving <math>R = \sqrt{ax^2+bx+c}</math>==

Assume (''ax''<sup>2</sup> + ''bx'' + ''c'') cannot be reduced to the following expression (''px'' + ''q'')<sup>2</sup> for some ''p'' and ''q''.

: <math>\int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln\left|2\sqrt{a}R+2ax+b\right| \qquad \mbox{(for }a>0\mbox{)}</math><!-- (4.1) [Abramowitz & Stegun p13 3.3.33] + verified by differentiation -->

: <math>\int\frac{dx}{R} = \frac{1}{\sqrt{a}}\,\operatorname{arsinh}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad \mbox{(for }a>0\mbox{, }4ac-b^2>0\mbox{)}</math><!-- (4.2) [Abramowitz & Stegun p13 3.3.34] + verified by differentiation -->

: <math>\int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln|2ax+b| \quad \mbox{(for }a>0\mbox{, }4ac-b^2=0\mbox{)}</math><!-- (4.3) [Abramowitz & Stegun p13 3.3.35] + verified by differentiation -->

: <math>\int\frac{dx}{R} = -\frac{1}{\sqrt{-a}}\arcsin\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad \mbox{(for }a<0\mbox{, }4ac-b^2<0\mbox{, }\left|2ax+b\right|<\sqrt{b^2-4ac}\mbox{)}</math><!-- (4.4) [Abramowitz & Stegun p13 3.3.36] + verified by differentiation -->

: <math>\int\frac{dx}{R^3} = \frac{4ax+2b}{(4ac-b^2)R}</math><!-- (4.5) [need reference] - verified by differentiation + special case of 4.7 below-->

: <math>\int\frac{dx}{R^5} = \frac{4ax+2b}{3(4ac-b^2)R}\left(\frac{1}{R^2}+\frac{8a}{4ac-b^2}\right)</math><!-- (4.6) [need reference] - verified by differentiation + special case of 4.7 below-->

: <math>\int\frac{dx}{R^{2n+1}} = \frac{2}{(2n-1)(4ac-b^2)}\left(\frac{2ax+b}{R^{2n-1}}+4a(n-1)\int\frac{dx}{R^{2n-1}}\right)</math><!-- (4.7) [need reference] - verified by differentiation only -->

: <math>\int\frac{x}{R}\;dx = \frac{R}{a}-\frac{b}{2a}\int\frac{dx}{R}</math><!-- (4.8) [Abramowitz & Stegun p13 3.3.39] + verified by differentiation -->

: <math>\int\frac{x}{R^3}\;dx = -\frac{2bx+4c}{(4ac-b^2)R}</math><!-- (4.9) [need reference] - verified by differentiation only -->

: <math>\int\frac{x}{R^{2n+1}}\;dx = -\frac{1}{(2n-1)aR^{2n-1}}-\frac{b}{2a}\int\frac{dx}{R^{2n+1}}</math><!-- (4.10) [need reference] - verified by differentiation only -->

: <math>\int\frac{dx}{xR}=-\frac{1}{\sqrt{c}}\ln\left(\frac{2\sqrt{c}R+bx+2c}{x}\right)</math><!-- (4.11) [Abramowitz & Stegun p13 implied by 3.3.38 + 3.3.33] + verified by differentiation -->

: <math>\int\frac{dx}{xR}=-\frac{1}{\sqrt{c}}\operatorname{arsinh}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right)</math><!-- (4.11) [Abramowitz & Stegun p13 implied by 3.3.38 + 3.3.34] + verified by differentiation -->

== Integrals involving <math>S = \sqrt{ax+b}</math>==
: <math>\int S {dx} = \frac{2 S^{3}}{3 a}</math>

: <math>\int \frac{dx}{S} = \frac{2S}{a}</math>

: <math>\int \frac{dx}{x S} =
\begin{cases}
-\frac{2}{\sqrt{b}} \mathrm{arcoth}\left( \frac{S}{\sqrt{b}}\right) & \mbox{(for }b > 0, \quad a x > 0\mbox{)} \\
-\frac{2}{\sqrt{b}} \mathrm{artanh}\left( \frac{S}{\sqrt{b}}\right) & \mbox{(for }b > 0, \quad a x < 0\mbox{)} \\
\frac{2}{\sqrt{-b}} \arctan\left( \frac{S}{\sqrt{-b}}\right) & \mbox{(for }b < 0\mbox{)} \\
\end{cases}</math>

: <math>\int\frac{S}{x}\,dx =
\begin{cases}
2 \left( S - \sqrt{b}\,\mathrm{arcoth}\left( \frac{S}{\sqrt{b}}\right)\right) & \mbox{(for }b > 0, \quad a x > 0\mbox{)} \\
2 \left( S - \sqrt{b}\,\mathrm{artanh}\left( \frac{S}{\sqrt{b}}\right)\right) & \mbox{(for }b > 0, \quad a x < 0\mbox{)} \\
2 \left( S - \sqrt{-b} \arctan\left( \frac{S}{\sqrt{-b}}\right)\right) & \mbox{(for }b < 0\mbox{)} \\
\end{cases}</math><!--yes it is S minus etc. in both cases as -->

: <math>\int \frac{x^{n}}{S} dx = \frac{2}{a (2 n + 1)} \left( x^{n} S - b n \int \frac{x^{n - 1}}{S} dx\right)</math><!-- (5.4) [need reference] - verified by differentiation only -->

: <math>\int x^{n} S dx = \frac{2}{a (2 n + 3)} \left(x^{n} S^{3} - n b \int x^{n - 1} S dx\right)</math><!-- (5.5) [need reference] - verified by differentiation only -->

: <math>\int \frac{1}{x^{n} S} dx = -\frac{1}{b (n - 1)} \left( \frac{S}{x^{n - 1}} + \left( n - \frac{3}{2}\right) a \int \frac{dx}{x^{n - 1} S}\right)</math>

== References ==
* {{cite book
|last=Peirce
|first=Benjamin Osgood
|title=A Short Table of Integrals
|origyear=1899
|edition=3rd revised ed.
|year=1929
|publisher=Ginn and Co.
|location=Boston
|pages=pp. 16-30
|chapter=Chap. 3
}}
* Milton Abramowitz and Irene A. Stegun, eds., ''[[Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables'' 1972, Dover: New York. ''(See [http://www.math.sfu.ca/~cbm/aands/page_13.htm chapter 3].)''

* S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)


{{Lists of integrals}}

[[Category:Integrals|Irrational functions]]
[[Category:Mathematics-related lists|Integrals of irrational functions]]

[[ar:قائمة تكاملات الدوال غير النسبية]]
[[bs:Spisak integrala iracionalnih funkcija]]
[[bg:Таблица с интеграли на ирационални функции]]
[[ca:Llista d'integrals de funcions irracionals]]
[[cs:Seznam integrálů iracionálních funkcí]]
[[es:Anexo:Integrales de funciones irracionales]]
[[eo:Listo de integraloj de malracionalaj funkcioj]]
[[fr:Primitives de fonctions irrationnelles]]
[[gl:Lista de integrais de funcións irracionais]]
[[hr:Popis integrala iracionalnih funkcija]]
[[id:Daftar integral dari fungsi irrasional]]
[[it:Tavola degli integrali indefiniti di funzioni irrazionali]]
[[km:តារាងអាំងតេក្រាលនៃអនុគមន៍អសនិទាន]]
[[pt:Anexo:Lista de integrais de funções irracionais]]
[[ro:Primitivele funcţiilor iraţionale]]
[[ru:Список интегралов от иррациональных функций]]
[[sk:Zoznam integrálov iracionálnych funkcií]]
[[sr:Списак интеграла ирационалних функција]]
[[sh:Popis integrala iracionalnih funkcija]]
[[tr:İrrasyonel fonksiyonların integralleri]]
[[zh:无理函数积分表]]

Revision as of 11:37, 10 October 2008

My edits and uploads to the Image namespace

  • Images I created
  • Composition of the United States 15.5 trillion US dollar CDS market at the end of 2008 Q2. Green tints indicate Prime asset CDSs, and red tints, reddish tints indicate sub-prime asset CDSs. Numbers followed by "Y" indicate the years until maturity.
    Composition of the United States 15.5 trillion US dollar CDS market at the end of 2008 Q2. Green tints indicate Prime asset CDSs, and red tints, reddish tints indicate sub-prime asset CDSs. Numbers followed by "Y" indicate the years until maturity.
  • Table render bug in Firefox 1 and 2
    Table render bug in Firefox 1 and 2
  • Overlapping bug of Mozilla browsers up to but not including Firefox 3.0.1
    Overlapping bug of Mozilla browsers up to but not including Firefox 3.0.1
  • Cropped image of Mozilla browser overlapping bug
    Cropped image of Mozilla browser overlapping bug
  • image affecting text flow
    image affecting text flow
  • images at right look better
    images at right look better
  • images disrupt flow
    images disrupt flow
  • images at right wide
    images at right wide
Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_integrals_of_irrational_functions&oldid=244351266"