# Homework Help: Differentiating an integral and finding f(x)

1. Dec 28, 2015

### supermiedos

1. The problem statement, all variables and given/known data
Find all f(x) satisfying:
∫f dx ∫1/f dx = -1

2. Relevant equations

3. The attempt at a solution

I solved for ∫1/f dx and differentiated both sides (using the quotient rule for the right side):
∫1/f dx = -1 / ∫f dx
1/f = f / (∫f dx)2
(∫f dx)2 = f2
∫fdx = ±f

f = ±f'

Solving the differential equation for f = f ' I get f = cex

But when I try to prove if my solution is correct, I got:

∫cex dx ∫ 1/(cex) dx = -1
(ex + k1)(-e-x + k2) = -1
And I don't know what to do to get -1 on the left side.

Could you give me a hint please?

2. Dec 28, 2015

### SammyS

Staff Emeritus
Wow! It's hard to see the ' on ƒ '
It's been edited. (Sorry for messing up that quote originally.)
It works if k1 = k2 = 0 .

Also, where did the constant, c, go in your last line?

3. Dec 28, 2015

### supermiedos

Solving the differential equation for f = f ' I get f = cex

But when I try to prove if my solution is correct, I got:

∫cex dx ∫ 1/(cex) dx = -1
(ex + k1)(-e-x + k2) = -1
And I don't know what to do to get -1 on the left side.

Could you give me a hint please?[/QUOTE]
It works is k1 = k2 = 0 .

But is it legal to do that? Giving values to fit the desired result?

Also, where did the constant, c, go in your last line?[/QUOTE]
"c" cancelled with the another "c" in the denominator of the second integral.

4. Dec 28, 2015

### SammyS

Staff Emeritus
c does not cancel if k1, k2 ≠ 0 .

You're right.

I was missing the fact that you can factor c out of the integrals or equivalently, you can have the integration constants "absorb" c.

Last edited: Dec 28, 2015