Question Number 222408 by Nicholas666 last updated on 25/Jun/25
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{1}} ^{\infty} \:\left(\frac{\mathrm{1}}{{x}}\right)^{\frac{{x}}{\:\sqrt{{x}−\mathrm{1}}}} \:{dx}\:=\:\:\:?? \\ $$$$ \\ $$
Answered by MrGaster last updated on 26/Jun/25
$$=\int_{\mathrm{0}} ^{\infty} \mathrm{exp}\left(−\left({t}+\frac{\mathrm{1}}{{t}}\right)\mathrm{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)\centerdot\mathrm{2}{tdt} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\infty} {t}\centerdot\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{−\left({t}+{t}^{−\mathrm{1}} \right)} {dt} \\ $$$$=\frac{\sqrt{\pi}}{\mathrm{2}}\mathrm{exp}\left(−\frac{\mathrm{1}}{\mathrm{4}}\right) \\ $$$$=\frac{\sqrt{\pi}}{\mathrm{2}}{e}^{−\frac{\mathrm{1}}{\mathrm{4}}} \\ $$