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If-f-x-x-0-e-f-t-dt-Determine-f-x-




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

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