Thanks a lot for that asianese!
The last part seems a bit confusing, so would it still be correct to see it this way?
F_0(y) = G_0(x) + C
\implies F_0(y) + C' = G_0(x) + C'',\;\text{where } C = C''-C'
\implies \int f(y) dy = \int g(x) dx
Comparing this to \frac{dy}{dx} = \frac{g(x)}{f(y)}...