Answer
$\ln |1+\sqrt x|+c$
Work Step by Step
Solve $\int\dfrac{dx}{2 \sqrt x+2x}$
or, $\int\dfrac{dx}{2 \sqrt x+2x}=\int\dfrac{dx}{2 \sqrt x(1+\sqrt x)}$
Let $p=1+\sqrt x$ and $dp=\dfrac{1}{2 \sqrt x }dx$
This implies $\int\dfrac{dx}{2 \sqrt x(1+\sqrt x)}=\int\dfrac{dp}{p}=\ln|p|+c$
Since, $p=1+\sqrt x$
Thus, $\int\dfrac{dx}{2 \sqrt x+2x}=\ln |1+\sqrt x|+c$
