Question Number 117841 by john santu last updated on 14/Oct/20
$$\int\:\mathrm{ln}\:\left(\mathrm{1}−{e}^{−\mathrm{2}{x}} \right)\:{dx}\:=? \\ $$
Answered by MJS_new last updated on 14/Oct/20
$$\int\mathrm{ln}\:\left(\mathrm{1}−\mathrm{e}^{−\mathrm{2}{x}} \right)\:{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{e}^{−\mathrm{2}{x}} \:\rightarrow\:{dx}=−\frac{\mathrm{e}^{\mathrm{2}{x}} }{\mathrm{2}}{dt}\right] \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{ln}\:\left(\mathrm{1}−{t}\right)}{{t}}{dt}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{Li}_{\mathrm{2}} \:{t}\:=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{Li}_{\mathrm{2}} \:\mathrm{e}^{−\mathrm{2}{x}} \:+{C} \\ $$
Commented by john santu last updated on 14/Oct/20
$${thank}\:{prof} \\ $$