PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (23)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] integration of fraction power expressions (Application)

The antiderivatives of every expression containing fraction powers can not be expressed by using elementary functions. However, there are types in which the integration succeeds after making a substitution.

  • $ \displaystyle\int R(x,\,x^{r_1},\,\ldots,\,x^{r_m})\,dx$, where $ R$ means a rational function of its arguments. If the common denominator of the fraction power exponents $ r_j$ is $ n$, the substitution
    $\displaystyle x = t^n, \quad dx = nt^{n-1}dt$
    changes each exponent to an integer and the whole integrand to a rational function in the variable $ t$.

    Example. For $ \displaystyle\int\frac{x^{\frac{1}{2}}}{x^{\frac{3}{4}}+1}\,dx$ the least common multiple of the denominators of $ \frac{1}{2}$ and $ \frac{3}{4}$ is 4, whence we make the substitution $ x = t^4$, $ dx = 4t^3dt$. Then we obtain

    $\displaystyle \int\frac{x^{\frac{1}{2}}}{x^{\frac{3}{4}}+1}\,dx = 4\!\int\frac{... ...{t^3+1}\right)dt = 4\left(\frac{t^3}{3}-\frac{1}{3}\ln\vert t^3+1\vert\right)+C$
    $\displaystyle = \frac{4}{3}\left(x^{\frac{3}{4}}-\ln\vert x^{\frac{3}{4}}+1\vert\right)+C.\\ $
  • In $ \displaystyle\int R\left(x,\,\left(\frac{ax+b}{cx+d}\right)^{r_1},\,\ldots,\,\left(\frac{ax+b}{cx+d}\right)^{r_m}\right)\,dx$, correspondently the substitution
    $\displaystyle \frac{ax+b}{cx+d} = t^n$
    changes the integrand to a rational function.

    Example. For $ \displaystyle\int\frac{\sqrt{x+4}}{x}\,dx$ we substitute $ x+4 = t^2$, $ dx = 2t\,dt$, getting

    $\displaystyle \int\frac{\sqrt{x+4}}{x}\,dx = 2\!\int\frac{t^2}{t^2-4}\,dt = 2\!... ...left(1+\frac{4}{t^2-4}\right)dt = 2t+2\ln\left\vert\frac{t-2}{t+2}\right\vert+C$
    $\displaystyle = 2\sqrt{x+4}+2\ln\left\vert\frac{\sqrt{x+4}-2}{\sqrt{x+4}+2}\right\vert+C.$

Bibliography

1
N. PISKUNOV: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele. Viies, täiendatud trükk. Kirjastus ``Valgus'', Tallinn (1965).



"integration of fraction power expressions" is owned by pahio.
(view preamble | get metadata)

View style:

See Also: fraction power, rational function, integration by substitution, substitution for integration


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: least common multiple, variable, integer, exponents, denominator, arguments, rational function, elementary functions, fraction powers, expression, antiderivatives
There is 1 reference to this entry.

This is version 3 of integration of fraction power expressions, born on 2008-02-22, modified 2008-02-24.
Object id is 10313, canonical name is IntegrationOfFractionPowerExpressions.
Accessed 764 times total.

Classification:
AMS MSC: 26A36 (Real functions :: Functions of one variable :: Antidifferentiation)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)