Question Number 60679 by maxmathsup by imad last updated on 24/May/19
$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{e}^{−{x}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}\:{dx} \\ $$
Commented by maxmathsup by imad last updated on 26/May/19
$${let}\:{A}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{1}+{e}^{−{x}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}\:{dx}\:\Rightarrow\mathrm{2}{A}\:=\:\int_{−\infty} ^{+\infty} \:\:\frac{{ln}\left(\mathrm{1}+{e}^{−{x}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}{dx} \\ $$$${let}\:\varphi\left({z}\right)\:=\frac{{ln}\left(\mathrm{1}+{e}^{−{z}^{\mathrm{2}} } \right)}{{z}^{\mathrm{2}} \:+\mathrm{4}}\:\:\:{we}\:{have}\:\varphi\left({z}\right)\:=\frac{{ln}\left(\mathrm{1}+{e}^{−{z}^{\mathrm{2}} } \right)}{\left({z}−\mathrm{2}{i}\right)\left({z}+\mathrm{2}{i}\right)}\:\:{so}\:{the}\:{poles}\:{of}\:\varphi\:{are}\: \\ $$$$\mathrm{2}{i}\:{and}\:−\mathrm{2}{i}\:\:{residus}\:{theorem}\:{give}\: \\ $$$$\int_{−\infty} ^{+\infty} \:\varphi\left({z}\right){dz}\:=\mathrm{2}{i}\pi{Res}\left(\varphi,\mathrm{2}{i}\right) \\ $$$${Res}\left(\varphi,\mathrm{2}{i}\right)\:={lim}_{{z}\rightarrow\mathrm{2}{i}} \left({z}−\mathrm{2}{i}\right)\varphi\left({z}\right)\:=\frac{{ln}\left(\mathrm{1}+{e}^{\left.−\left(\mathrm{2}{i}\right)^{\mathrm{2}} \right)} \right.}{\mathrm{4}{i}}\:=\frac{{ln}\left(\mathrm{1}+{e}^{\mathrm{4}} \right)}{\mathrm{4}{i}}\:\Rightarrow \\ $$$$\int_{−\infty} ^{+\infty} \:\varphi\left({z}\right){dz}\:=\mathrm{2}{i}\pi\:\frac{{ln}\left(\mathrm{1}+{e}^{\mathrm{4}} \right)}{\mathrm{4}{i}}\:=\frac{\pi}{\mathrm{2}}\:{ln}\left(\mathrm{1}+{e}^{\mathrm{4}} \right)\:\Rightarrow \\ $$$${A}\:=\frac{\pi}{\mathrm{4}}{ln}\left(\mathrm{1}+{e}^{\mathrm{4}} \right) \\ $$