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

$$Findallfunctionsthatsatisfythe$$$$equation$$$$[\int f\left(x\right)dx][\int \frac{1}{f\left(x\right)}dx]=-1$$

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

$$\int \frac{1}{f\left(x\right)}dx=-\frac{1}{\int f\left(x\right)dx}$$$$\frac{1}{f\left(x\right)}=\frac{f\left(x\right)}{{(\int f\left(x\right)dx)}^2}$$$$\Rightarrow \int f\left(x\right)dx=\pm f\left(x\right)$$$$\Rightarrow f\left(x\right)=\pm f^{\prime }\left(x\right)$$$$\int \frac{df\left(x\right)}{f\left(x\right)}=\pm \int dx$$$$\ln f\left(x\right)=\pm x+c$$$$\Rightarrow f\left(x\right)={Ce}^{\pm x}$$$$$$$$check:$$$$\int f\left(x\right)dx=\pm {Ce}^{\pm x}$$$$\int \frac{1}{f\left(x\right)}dx=\pm \int \frac{1}{C}{e}^{\mp x}dx=\mp \frac{1}{C}{e}^{\mp x}$$$$[\int f\left(x\right)dx][\int \frac{1}{f\left(x\right)}dx]=(\pm {Ce}^{\pm x})(\mp \frac{1}{C}{e}^{\pm x})=-1$$

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

$$thankssir$$

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