"You have 1/x dx/dy = 1/y, and you're assuming that x is a function of y. From this you can conclude that an antiderivative of the left hand side of the above differs from an antiderivative of the righthand side by a constant.
This is saying: ∫1/x dx/dy dy=∫ 1/y dy. But, by the chain rule, you can write the left hand side as ∫1/x dx. The method of "separating variables" leads you, safely, to this conclusion."
Can you please explain how you used the chain rule to get ∫1/x dx from ∫1/x dx/dy dy?
Thank you
Let $F(x)$ be an antiderivative of $1/x$ with respect to $x$ so that $F'(x) = 1/x$ and $F(x) = \int 1/x \, dx$. Then by the chain rule, notice that: $$ \frac{d}{dy} F(x) = \frac{1}{x}\frac{dx}{dy} $$ Hence, it follows that: $$ \int \frac{1}{x} \, \frac{dx}{dy} \, dy = \int \frac{d}{dy} F(x) \, dy = F(x) + C = \int \frac{1}{x} \, dx $$ as desired.