(adsbygoogle = window.adsbygoogle || []).push({}); The problem:

[tex](\int f(x)dx)(\int \frac{1}{f(x)} dx) = -1[/tex]

solve for f(x)

The attempt at a solution

divide both sides by integral(f(x)dx):

[tex]\int \frac{1}{f(x)} dx = \frac{-1}{\int f(x)dx}[/tex]

took derivative of both sides:

[tex]\frac{1}{f(x)} = \frac{f(x)}{(\int f(x) dx)^2}[/tex]

multiplied both sides by f(x):

[tex]1 = \frac{f(x)^2}{(\int f(x) dx)^2}[/tex]

square root:

[tex]1 = \frac{f(x)}{\int f(x) dx}[/tex]

Now, here I applied L'Hopital's incorrectly (this isn't even a limit I know!) by taking derivative of top and bottom on the right side; it seemed to work later that's why I tried to stick with it:

[tex]1 = \frac{f'(x)}{f(x)}[/tex]

took antiderivative of both sides...

[tex]x = ln|f(x)|[/tex]

[tex]e^x = |f(x)|[/tex]

[tex]f(x) = e^x and -e^x[/tex]

when plugging those two answers into the original equation it seems to work...

but something isn't right and anyway there is no way that only those two functions satisfy that. help?

(first time trying to use the tex tags so i have my fingers crossed that I did not make a mistake in formatting..)

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# Homework Help: What are all the functions that satisfy (Calculus II)

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