Humorous integration example? I was just reading though an introductory calculus book and it has the note:

NOTE When integrating quotients, do not integrate the numerator and denominator separately. This is no more valid in integration than it is in differentiation.

Now that's fair enough to point out and it gives a nice example too. But out of curiosity...
Are there examples of functions $f ,g: \Bbb R \rightarrow \Bbb R $ whereby:
$\int \frac {f}{g} =\frac {\int f}{\int g}$.
Say for clarity you have a choice of the constants in the  antiderivatives, and $f \not\equiv 0$.
I imagine it might possibly be easier if you choose definite integrals, just none spring to mind! 
Maybe there's a link to a similar question on here? 

It is easy to verify that, in the version suggested by @MPW, for $f=e^{ax}$, $g=e^{bx}$, we have
$$
a=\frac{b}{b-1}.
$$
For $b=2$ we have the answer obtained by Sam.
Edit:
There is a sketch, that this is near general solution.
We denote integral by $F$, as it is up to constant some  function. Then we have
$$
F(fg)=F(f)F(g).
$$
It is known, that the solution (up to constants again) is  $F(g)=g^a$, $a\in\bf R$.
It means that $g(x)=ag^{a-1}(x)g'(x)$. It is a Bernoulli equation, with solutions of the form $Ce^{bx}$.
