Find-all-functions-that-satisfy-the-equation-f-x-dx-1-f-x-dx-1-

Question Number 86734 by Tony Lin last updated on 30/Mar/20

F i n d a l l f u n c t i o n s t h a t s a t i s f y t h e

e q u a t i o n

[\int f (x) d x] [\int \frac{1}{f (x)} d x] = - 1

Answered by mr W last updated on 30/Mar/20

\int \frac{1}{f (x)} d x = - \frac{1}{\int f (x) d x}

\frac{1}{f (x)} = \frac{f (x)}{{(\int f (x) d x)}^{2}}

\Rightarrow \int f (x) d x = \pm f (x)

\Rightarrow f (x) = \pm f^{'} (x)

\int \frac{d f (x)}{f (x)} = \pm \int d x

\ln f (x) = \pm x + c

\Rightarrow f (x) = {C e}^{\pm x}

c h e c k :

\int f (x) d x = \pm {C e}^{\pm x}

\int \frac{1}{f (x)} d x = \pm \int \frac{1}{C} e^{\mp x} d x = \mp \frac{1}{C} e^{\mp x}

[\int f (x) d x] [\int \frac{1}{f (x)} d x] = (\pm {C e}^{\pm x}) (\mp \frac{1}{C} e^{\pm x}) = - 1

Commented by Tony Lin last updated on 30/Mar/20

t h a n k s s i r