If $g(x)=∫\frac{dx}{x^{\frac{1}{2}}+x^{\frac{1}{6}}}$, then $g(1) - g(0)$ is:

$5+6log_e2$

Taking LCM of denominator of $\frac{1}{2}$ and $\frac{1}{6}$

which is 6

So $x = t^6$

$⇒ dx = 6t^5$

$g(x)=\int\frac{6t^5}{t^3+t}dt⇒\int\frac{6t^5}{t(t^2+t)}dt$

$⇒t\frac{6t^4}{(t^2+t)}dt⇒6\int\frac{(t^3+1)-1}{t^2+1}dt⇒6\int t^2-t+1-\frac{1}{t^2+1}$