Question Number 36799 by abdo.msup.com last updated on 05/Jun/18
$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{{t}} {ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{t}} \right){dt}\:. \\ $$
Commented by math khazana by abdo last updated on 11/Jun/18
$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{{t}} \:{ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{t}} \right){dt}\:{by}\:{parts} \\ $$$${u}^{'} ={e}^{{t}} \:\:{and}\:{v}\:={ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{t}} \right)\:\Rightarrow \\ $$$${I}\:=\:\left[\:{e}^{{t}} {ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{t}} \right)\right]_{\mathrm{0}} ^{+\infty} \:−\int_{\mathrm{0}} ^{\infty} \:{e}^{{t}} \:\:\frac{−\mathrm{2}\:{e}^{−\mathrm{2}{t}} }{\mathrm{1}+{e}^{−\mathrm{2}{t}} }{dt} \\ $$$$=−{ln}\left(\mathrm{2}\right)\:+\mathrm{2}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−{t}} }{\mathrm{1}+{e}^{−\mathrm{2}{t}} }\:{dt}\:{but} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−{t}} }{\mathrm{1}+{e}^{−\mathrm{2}{t}} }\:{dt}\:=_{{e}^{{t}} ={x}} \:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\:\frac{\mathrm{1}}{{x}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)}\:\frac{{dx}}{{x}} \\ $$$$=\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:+\mathrm{1}}\:=\left[{arctanx}\right]_{\mathrm{1}} ^{+\infty} =\frac{\pi}{\mathrm{2}}\:−\frac{\pi}{\mathrm{4}}\:=\frac{\pi}{\mathrm{4}}\:\Rightarrow \\ $$$$\bigstar{I}\:=\:−{ln}\left(\mathrm{2}\right)\:+\frac{\pi}{\mathrm{2}}\:\bigstar \\ $$
Answered by MJS last updated on 06/Jun/18
$$\int\mathrm{e}^{{t}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{−\mathrm{2}{t}} \right){dt}= \\ $$$$\:\:\:\:\:\:\:\:\:\:\begin{bmatrix}{\int{f}'{g}={fg}−\int{fg}'}\\{{f}'=\mathrm{e}^{{t}} \:\Rightarrow\:{f}=\mathrm{e}^{{t}} }\\{{g}=\mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{−\mathrm{2}{t}} \right)\:\Rightarrow\:{g}'=−\frac{\mathrm{2e}^{−\mathrm{2}{t}} }{\mathrm{1}+\mathrm{e}^{−\mathrm{2}{t}} }}\end{bmatrix} \\ $$$$=\mathrm{e}^{{t}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{−\mathrm{2}{t}} \right)−\mathrm{2}\int−\frac{\mathrm{e}^{−{t}} }{\mathrm{1}+\mathrm{e}^{−\mathrm{2}{t}} }{dt}= \\ $$$$\:\:\:\:\:\:\:\:\:\:\left[{u}=−\mathrm{e}^{{t}} \:\rightarrow\:{dt}=−\mathrm{e}^{{t}} {du}\right] \\ $$$$=\mathrm{e}^{{t}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{−\mathrm{2}{t}} \right)−\mathrm{2}\int\frac{{u}}{\mathrm{1}+{u}^{\mathrm{2}} }{du}= \\ $$$$=\mathrm{e}^{{t}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{−\mathrm{2}{t}} \right)−\mathrm{2arctan}\:{u}= \\ $$$$=\mathrm{e}^{{t}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{−\mathrm{2}{t}} \right)−\mathrm{2arctan}\:−\mathrm{e}^{{t}} = \\ $$$$=\mathrm{e}^{{t}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{−\mathrm{2}{t}} \right)+\mathrm{2arctan}\:\mathrm{e}^{{t}} +{C} \\ $$$$ \\ $$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\mathrm{e}^{{t}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{−\mathrm{2}{t}} \right){dt}=\frac{\pi}{\mathrm{2}}−\mathrm{ln}\:\mathrm{2} \\ $$