Question Number 38127 by maxmathsup by imad last updated on 22/Jun/18
$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} \sqrt{\mathrm{1}+{e}^{−\mathrm{2}{x}} }{dx} \\ $$
Commented by abdo mathsup 649 cc last updated on 23/Jun/18
$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} \sqrt{\mathrm{1}+{e}^{−\mathrm{2}{x}} }{dx}\:{changement}\:{e}^{−{x}} ={t} \\ $$$${give}\:\:−{x}={ln}\left({t}\right)\:\Rightarrow{x}=−{ln}\left({t}\right)\:{and} \\ $$$${I}\:=\:−\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{t}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:\frac{−{dt}}{{t}}\: \\ $$$$=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:\:{also}\:{chsng}.\:{t}={shu}\:{give} \\ $$$${I}\:=\:\int_{\mathrm{0}} ^{{argsh}\left(\mathrm{1}\right)} \:\:{ch}\left({u}\right){ch}\left({u}\right){du} \\ $$$$=\:\int_{\mathrm{0}} ^{{ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)} \:\frac{\mathrm{1}+{ch}\left(\mathrm{2}{u}\right)}{\mathrm{2}}{du} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)\:\:+\frac{\mathrm{1}}{\mathrm{4}}\left[{sh}\left(\mathrm{2}{u}\right)\right]_{\mathrm{0}} ^{{ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)\:\:+\frac{\mathrm{1}}{\mathrm{4}}\left[\:\frac{{e}^{\mathrm{2}{u}} \:−{e}^{−\mathrm{2}{u}} }{\mathrm{2}}\right]_{\mathrm{0}} ^{{ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)\:+\frac{\mathrm{1}}{\mathrm{8}}\left\{\:\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)^{\mathrm{2}} \:−\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)^{−\mathrm{2}} \right\} \\ $$