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Question-70909




Question Number 70909 by rajesh4661kumar@gmail.com last updated on 09/Oct/19
Commented by rajesh4661kumar@gmail.com last updated on 10/Oct/19
28
$$\mathrm{28} \\ $$
Commented by Abdo msup. last updated on 10/Oct/19
f(t)=∫_0 ^1  ((x^t −1)/(lnx))dx  changement lnx=−u give  f(t)=−∫_0 ^∞    ((e^(−tu) −1)/(−u))(−e^(−u) )du  =−∫_0 ^∞   ((e^(−(t+1)u) −e^(−u) )/u) du=∫_0 ^∞ ((e^(−u) −e^(−(t+1)u) )/u)du  f^′ (t) = ∫_0 ^∞    e^(−(t+1)u) du =[−(1/(t+1))e^(−(t+1)u) ]_(u=0) ^∞   =(1/(t+1)) ⇒f(t)=ln(t+1) +c  f(o)=0=c ⇒f(t)=ln(t+1)   (t≥0)
$${f}\left({t}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{x}^{{t}} −\mathrm{1}}{{lnx}}{dx}\:\:{changement}\:{lnx}=−{u}\:{give} \\ $$$${f}\left({t}\right)=−\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−{tu}} −\mathrm{1}}{−{u}}\left(−{e}^{−{u}} \right){du} \\ $$$$=−\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−\left({t}+\mathrm{1}\right){u}} −{e}^{−{u}} }{{u}}\:{du}=\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{u}} −{e}^{−\left({t}+\mathrm{1}\right){u}} }{{u}}{du} \\ $$$${f}^{'} \left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{−\left({t}+\mathrm{1}\right){u}} {du}\:=\left[−\frac{\mathrm{1}}{{t}+\mathrm{1}}{e}^{−\left({t}+\mathrm{1}\right){u}} \right]_{{u}=\mathrm{0}} ^{\infty} \\ $$$$=\frac{\mathrm{1}}{{t}+\mathrm{1}}\:\Rightarrow{f}\left({t}\right)={ln}\left({t}+\mathrm{1}\right)\:+{c} \\ $$$${f}\left({o}\right)=\mathrm{0}={c}\:\Rightarrow{f}\left({t}\right)={ln}\left({t}+\mathrm{1}\right)\:\:\:\left({t}\geqslant\mathrm{0}\right) \\ $$
Answered by Henri Boucatchou last updated on 10/Oct/19
What  question(s) ?
$${What}\:\:{question}\left({s}\right)\:? \\ $$

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