Integrate: x^(1/2) / x^(1/2) -x^(1/3)

Solution:

$I=\int \frac{x^\frac{1}{2}}{x^\frac{1}{2}-x^\frac{1}{3}}dx$   L.C.M  of $2$ and $3$ . hence  we put

$z^{6}=x,$   then  $6z^{5}dz=dx,x^\frac{1}{2}=z^\frac{6}{2}=z^{3},$

$\Rightarrow$      $I=6\int [z^{5}+z^{4}+z^{3}+z^{2}+z+1+\frac{1}{z-1}]dz$

$\Rightarrow$      $I=6[\frac{z^{6}}{6}+\frac{z^{5}}{5}+\frac{z^{4}}{4}+\frac{z^{3}}{2}+\frac{z^{2}}{2}+z+\log (z-1)]+c$

$\Rightarrow$         $I=6[\frac{x}{6}+\frac{x^\frac{5}{6}}{5}+\frac{x^\frac{2}{3}}{4}+\frac{x^\frac{1}{2}}{3}+\frac{x^\frac{1}{3}}{3}+x^\frac{1}{6}+\log (x^\frac{1}{6}-1)]+c.$

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