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Question Number 199511 by SANOGO last updated on 04/Nov/23

∫_1 ^(+oo) ln(1+lnx)^(−lnx)  dx

$$\int_{\mathrm{1}} ^{+{oo}} {ln}\left(\mathrm{1}+{lnx}\right)^{−{lnx}} \:{dx} \\ $$

Commented by SANOGO last updated on 04/Nov/23

thank you

$${thank}\:{you} \\ $$

Answered by witcher3 last updated on 04/Nov/23

diverge  ln(1+ln(x))^(−ln(x)) =ln((1/((1+ln(x)^(ln(x)) )))  ∀x≥e  ln((1/((1+ln(x))^(ln(x)) )))≤ln((1/2^2 ))≤−ln(4)  diverge −∞

$$\mathrm{diverge} \\ $$$$\mathrm{ln}\left(\mathrm{1}+\mathrm{ln}\left(\mathrm{x}\right)\right)^{−\mathrm{ln}\left(\mathrm{x}\right)} =\mathrm{ln}\left(\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ln}\left(\mathrm{x}\right)^{\mathrm{ln}\left(\mathrm{x}\right)} \right.}\right) \\ $$$$\forall\mathrm{x}\geqslant\mathrm{e} \\ $$$$\mathrm{ln}\left(\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ln}\left(\mathrm{x}\right)\right)^{\mathrm{ln}\left(\mathrm{x}\right)} }\right)\leqslant\mathrm{ln}\left(\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\right)\leqslant−\mathrm{ln}\left(\mathrm{4}\right) \\ $$$$\mathrm{diverge}\:−\infty \\ $$

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