integration of fraction power expressions

The antiderivatives of every expression containing fraction powers can not be expressed by using elementary functionsMathworldPlanetmath. However, there are after making a substitution.

  • R(x,xr1,,xrm)𝑑x,  where R means a rational function of its arguments. If the common denominator of the fraction power exponentsMathworldPlanetmath rj is n, the substitution


    changes each exponent to an integer and the whole integrand to a rational function in the variable t.

    Example.  For  x12x34+1𝑑x  the least common multiple of the denominators of 12 and 34 is 4, whence we make the substitution  x=t4,  dx=4t3dt.  Then we obtain

    x12x34+1𝑑x= 4t5dtt3+1= 4(t2-t2t3+1)𝑑t= 4(t33-13ln|t3+1|)+C
  • In R(x,(ax+bcx+d)r1,,(ax+bcx+d)rm)𝑑x,  correspondently the substitution


    changes the integrand to a rational function.

    Example.  For  x+4x𝑑x  we substitute  x+4=t2,  dx=2tdt,  getting

    x+4x𝑑x= 2t2t2-4𝑑t= 2(1+4t2-4)𝑑t= 2t+2ln|t-2t+2|+C
    = 2x+4+2ln|x+4-2x+4+2|+C.


  • 1 N. Piskunov: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele. Viies, täiendatud trükk.  Kirjastus “Valgus”, Tallinn (1965).
Title integration of fraction power expressions
Canonical name IntegrationOfFractionPowerExpressions
Date of creation 2013-03-22 17:50:33
Last modified on 2013-03-22 17:50:33
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Application
Classification msc 26A36
Related topic FractionPower
Related topic RationalFunction
Related topic IntegrationBySubstitution
Related topic SubstitutionForIntegration