Question

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

Answer 1

We have,

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

LCM of $2and3$ $=6$

Let,

$x=t^6$

$\frac{dx}{dt}=6t^5$

$dx=6t^{5}dt$

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

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

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

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

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

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

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

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

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

Put $t=\sqrt[6]{6x}$

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

Hence, this is the answer.