TITLE: $\int f(x) dx\cdot \int \frac{1}{f(x)}dx = c$ where $c$ is a constant. Find $f(x)$
QUESTION [5 upvotes]: $$\int f(x) dx\cdot \int \frac{1}{f(x)}dx = c$$ where $c$ is a constant. Find $f(x)$. 


Off first glance it seems that $f(x)$ is some form of $e^x$, but how does one go about doing this analytically (only 1 mark for guessing). I took the derivative of this but the resulting function was still ugly. I thought maybe treating it as a first order linear differential equation would work, but I wasn't sure how to put it in the form $y' + P(x)y = Q(x)$

REPLY [7 votes]: Let $F(x)$ be the antiderivative of $f(x)$.
$$\int\frac1{F'(x)}dx=\frac c{F(x)},$$
$$\frac1{F'(x)}=-\frac{cF'(x)}{F^2(x)}.$$
Then
$$\left(F'(x)+\frac1{\sqrt{-c}}F(x)\right)\left(\,F'(x)-\frac1{\sqrt{-c}}F(x)\right)=0,$$
has the desired form (and of course the solutions are exponential). There are no real solutions for $c\ge0$.