integration of fraction power expressions

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

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$\displaystyle\int R(x,\,x^{r_{1}},\,\ldots,\,x^{r_{m}})\,dx$ , where $R$ means a rational function of its arguments. If the common denominator of the fraction power exponents $r_{j}$ is $n$ , the substitution

$x\;:=\;t^{n},\qquad dx\;=\;nt^{n-1}dt$

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 $x=t^{4}$ , $dx=4t^{3}dt$ . Then we obtain

$\int\frac{x^{\frac{1}{2}}}{x^{\frac{3}{4}}+1}\,dx\;=\;4\!\int\frac{t^{5}dt}{t^% {3}+1}\;=\;4\!\int\left(t^{2}-\frac{t^{2}}{t^{3}+1}\right)dt\;=\;4\left(\frac{% t^{3}}{3}-\frac{1}{3}\ln|t^{3}+1|\right)+C$

$=\;\frac{4}{3}\left(x^{\frac{3}{4}}-\ln|x^{\frac{3}{4}}+1|\right)+C.\\$

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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

$\frac{ax+b}{cx+d}\;:=\;t^{n}$

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

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

$=\;2\sqrt{x+4}+2\ln\left|\frac{\sqrt{x+4}-2}{\sqrt{x+4}+2}\right|+C.$

References

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