# ODE's, Find all functions f that help satisfy the equation

## Homework Statement

Find all functions f that help satisfy the equation
$\left(\int f(x)dx\right)\left(\int\frac{1}{f(x)}dx\right) = -1$

## The Attempt at a Solution

I'm not quite sure what to do here. When I differentiate I get $f(x)\int1/ f(x) dx + 1/f(x)\int f(x) dx$ which doesn't seem to help if I keep differentiating. I'm kinda stuck on this problem so any help is appreciated. Thanks

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I've been looking at this and trying to figure it out.

At first, I was thinking of a trig function, but after more thinking that seems to be out. Now I'm thinking something like $f(x) = e^x$, though I haven't really tried to work it out past that part, just an idea for you.

The answer clearly seems to be f(x) = e**x I just can't get to proving it.

That's the main trouble I was having. I get how to prove that it's not other functions by solving $\int x^n * \int \frac{1}{x^n}$, which shows that a standard function to degree n will always result in something such as $\frac{-x^{n+1}}{n+1}$, if $|n|>1$. Knowing this shows that you must have a function that repeats itself when integrated, such as a trig function or $e^x$. This is the closest thing that I've gotten to so far, but it still isn't much of a mathematical proof at all.