ln integration (differential equations problem)

Question

I'm trying to solve

$$\frac{(\sqrt x + x)\,dy}{dx} = \sqrt y + y$$

I can separate the variables and get

$$\frac {dy} {\sqrt y + y} = \frac{dx}{\sqrt x + x}$$

I know that integrating $$\frac {1}{x} = \ln (x)$$ but in this case why can't I just say that the integral of the above with reference to $x$ is $\ln$ (denominator)?

Answer 1

Hint

By changing the variable you get:

$$\int\frac{dx}{x+\sqrt{x}}\underbrace{=}_{x=t^2} \int\frac{2dt}{t+1}=2\ln |t+1|+c\underbrace{=}_{x=t^2}2\ln (\sqrt{x}+1) +c.$$