Question

Solve
$\int \frac{dx}{(x^{1/2}+x^{1/3})}$

Answer 1

$I=\int \frac{dx}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$

Put $x=t^6$

$dx=6t^{3}dt$

$=\int \frac{6t^{5}dt}{t^3+t^2}=\int \frac{6t^3}{t+1}dt$

$=6\int (\frac{t^3+1}{t+1}-\frac{1}{t+1})dt=6\int \left(\right(t^2-t+1)dt-\frac{1}{t+1})dt$

$=6\int \left[\right(t^2-t+1)dt-\frac{1}{t+1}dt]$

$6[\frac{t^3}{3}-\frac{t^2}{2}+t-\ln (t+1\left)\right]+C$

$=2x^{\frac{1}{2}}-3x^{\frac{1}{3}}-6\log (x^{\frac{1}{6}}+1)+C$