# 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.

• $\displaystyle\int R(x,\,x^{r_{1}},\,\ldots,\,x^{r_{m}})\,dx$,  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

 $\int\frac{x^{\frac{1}{2}}}{x^{\frac{3}{4}}+1}\,dx\;=\;4\!\int\frac{t^{5}dt}{t^% {3}+1}\;=\;4\!\int\left(t^{2}-\frac{t^{2}}{t^{3}+1}\right)dt\;=\;4\left(\frac{% t^{3}}{3}-\frac{1}{3}\ln|t^{3}+1|\right)+C$
 $=\;\frac{4}{3}\left(x^{\frac{3}{4}}-\ln|x^{\frac{3}{4}}+1|\right)+C.\\$
• 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 IntegrationOfFractionPowerExpressions 2013-03-22 17:50:33 2013-03-22 17:50:33 pahio (2872) pahio (2872) 7 pahio (2872) Application msc 26A36 FractionPower RationalFunction IntegrationBySubstitution SubstitutionForIntegration