Hi guys, I have a question about whether it's valid to integrate both sides of an equation. The question is asking about the relationship between two functions $\displaystyle f,g$ such that $\displaystyle (f/g)'=f'/g'$. We're supposed to express $\displaystyle f$ it terms of $\displaystyle g$ and $\displaystyle g'$. So after doing some work, I got

$\displaystyle \frac{f'}{f}=\frac{(g')^2}{g(g'-g)}$

Now I'd like to integrate both sides to get something like

$\displaystyle \ln f(x)=\int_a^x\frac{(g'(y))^2}{g(y)(g(y)'-g(y))}dy$

which I can then exponentiate to get an expression for $\displaystyle f$. My question is, is this valid? Are my limits of integration correct? Would I be better off just writing

$\displaystyle \ln \circ f=\int\frac{(g')^2}{g(g'-g)}$

and just keep it all in terms of functions?

Thanks,

mtdim