Is there a reason why $\ln (x^2 + x) - \ln x = \ln(x+1)$ would not always be true? I ask because wolfram alpha doesn't just say that it is true, seems that there's some condition that I'm missing

solved! see answers here: math.stackexchange.com/questions/4181629/…

@user21820 hey! sorry for bothering you again, but I was trying out your technique for solving "separable" equations

and I got stuck on this problem $\frac{1}{x} = \frac{1}{y} \frac{dy}{dx}$

using the two theorems you showed me, I got to $\int \frac{1}{y} \text{dy} = \int \frac{1}{x} \text{dx} + C$

but now, to go forward, is the correct approach: $y = e^{\ln x}e^C$?

(sorry for bothering you again on the same question)

to this previous query: I wrote $y = xk$ and moved onwards with the problem, I just had never encountered an unknown constants that's multiplied instead of added after an integral, but it works just fine