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find-f-t-0-e-x-ln-1-e-tx-dx-with-t-gt-0-2-let-u-n-0-e-x-ln-1-e-nx-dx-find-lim-n-u-n-




Question Number 37243 by math khazana by abdo last updated on 11/Jun/18
find f (t) =∫_0 ^∞  e^x ln(1+e^(−tx) )dx with t >0 .  2) let u_n = ∫_0 ^∞   e^x ln(1+e^(−nx) ) dx  find lim_(n→+∞) u_n  .
$${find}\:{f}\:\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{e}^{{x}} {ln}\left(\mathrm{1}+{e}^{−{tx}} \right){dx}\:{with}\:{t}\:>\mathrm{0}\:. \\ $$$$\left.\mathrm{2}\right)\:{let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{{x}} {ln}\left(\mathrm{1}+{e}^{−{nx}} \right)\:{dx} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \:. \\ $$

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