Question Number 196394 by Erico last updated on 24/Aug/23
$$\mathrm{If}\:\:{f}\left({x}\right)=\underset{\:\mathrm{0}} {\int}^{\:{x}} {e}^{−{f}\left({t}\right)} {dt} \\ $$$$\mathrm{Determine}\:{f}\left({x}\right) \\ $$
Answered by mr W last updated on 24/Aug/23
$$\frac{{df}\left({x}\right)}{{dx}}={e}^{−{f}\left({x}\right)} \\ $$$${e}^{{f}\left({x}\right)} {df}\left({x}\right)={dx} \\ $$$$\int{e}^{{f}\left({x}\right)} {df}\left({x}\right)=\int{dx} \\ $$$${e}^{{f}\left({x}\right)} ={x}+{C} \\ $$$$\Rightarrow{f}\left({x}\right)=\mathrm{ln}\:\left({x}+{C}\right) \\ $$