Question Number 77752 by abdomathmax last updated on 09/Jan/20
$${let}\:{f}\left(\lambda\right)\:=\int_{−\infty} ^{+\infty} \:\frac{{sin}\left(\:\lambda{e}^{{x}} \:+{e}^{−{x}} \right)}{{x}^{\mathrm{2}} \:+\lambda^{\mathrm{2}} }{dx}\:{with}\:\lambda\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{detdrmine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left(\lambda\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left(\lambda\right)\:{at}\:{form}\:{ofintergral}\:{and}\:{find} \\ $$$${its}\:{value}. \\ $$
Commented by mathmax by abdo last updated on 10/Jan/20
$${f}\left(\lambda\right)=\int_{−\infty} ^{+\infty} \:\frac{{sin}\left(\lambda{e}^{{x}} \:+{e}^{−{x}} \right)}{{x}^{\mathrm{2}} \:+\lambda^{\mathrm{2}} }\:\Rightarrow{f}\left(\lambda\right)=_{{x}=\lambda{t}} \:\:\int_{−\infty} ^{+\infty} \:\frac{{sin}\left(\lambda\:{e}^{\lambda{t}} \:+{e}^{−\lambda{t}} \right)}{\lambda^{\mathrm{2}} \left({t}^{\mathrm{2}} \:+\mathrm{1}\right)}\lambda\left({dt}\right) \\ $$$$=\frac{\mathrm{1}}{\lambda}\:\int_{−\infty} ^{+\infty} \:\frac{{sin}\left(\lambda\:{e}^{\lambda{t}} \:+{e}^{−\lambda{t}} \right)}{{t}^{\mathrm{2}} \:+\mathrm{1}}{dt}\:\:\:\:\left(\:\:{we}\:{take}\:\lambda>\mathrm{0}\right)\:\Rightarrow \\ $$$$\lambda{f}\left(\lambda\right)\:={Im}\left(\int_{−\infty} ^{+\infty} \:\frac{{e}^{{i}\left(\:\lambda{e}^{\lambda{t}} +{e}^{−\lambda{t}} \right)} }{{t}^{\mathrm{2}} \:+\mathrm{1}}{dt}\right){let}\:{W}\left({z}\right)=\frac{{e}^{{i}\left(\lambda\:{e}^{\lambda{z}} \:+{e}^{−\lambda{z}} \right)} }{{z}^{\mathrm{2}} \:+\mathrm{1}} \\ $$$$\Rightarrow{W}\left({z}\right)\:=\frac{{e}^{{i}\left(\lambda\:{e}^{\lambda{z}} \:+{e}^{−\lambda{z}} \right)} }{\left({z}−{i}\right)\left({z}+{i}\right)}\:{and}\:\int_{−\infty} ^{+\infty} \:{W}\left({z}\right){dz}\:=\mathrm{2}{i}\pi\:{Res}\left({W},{i}\right) \\ $$$$=\mathrm{2}{i}\pi\:×\frac{{e}^{{i}\left(\lambda\:{e}^{{i}\lambda} \:+{e}^{−{i}\lambda} \right)} }{\mathrm{2}{i}}\:\:{we}\:{have}\: \\ $$$${i}\left(\lambda\:{e}^{{i}\lambda} \:+{e}^{−{i}\lambda} \right)\:={i}\left(\lambda\:{cos}\left(\lambda\right)+\lambda{i}\:{sin}\left(\lambda\right)\:+{cos}\left(\lambda\right)−{isin}\left(\lambda\right)\right) \\ $$$$={i}\:\lambda\:{cos}\left(\lambda\right)−\lambda{sin}\left(\lambda\right)\:+{icos}\left(\lambda\right)\:+{sin}\lambda \\ $$$$=\left(\mathrm{1}−\lambda\right){sin}\lambda\:+{i}\left(\mathrm{1}+\lambda\right){cos}\left(\lambda\right)\:\Rightarrow \\ $$$${e}^{{i}\left(\lambda\:{e}^{{i}\lambda} \:+{e}^{\left.−{i}\lambda\right)} \right.} \:={e}^{\left(\mathrm{1}−\lambda\right){sin}\left(\lambda\right)} \left\{\:{cos}\left(\mathrm{1}+\lambda\right){cos}\lambda\:+{isin}\left(\mathrm{1}+\lambda\right){cos}\lambda\right\}\:\Rightarrow \\ $$$$\int_{−\infty} ^{+\infty} \:{W}\left({z}\right){dz}\:=\pi\:{e}^{\left(\mathrm{1}−\lambda\right){sin}\lambda} \:{cos}\left(\left(\mathrm{1}+\lambda\right){cos}\lambda\right)\:+{i}\:\pi\:{e}^{\left(\mathrm{1}−\lambda\right){sin}\lambda} {sin}\left(\left(\mathrm{1}+\lambda\right){cos}\lambda\right) \\ $$$$\Rightarrow\lambda\:{f}\left(\lambda\right)\:=\pi\:{e}^{\left(\mathrm{1}−\lambda\right){sin}\lambda} \:{sin}\left\{\left(\mathrm{1}+\lambda\right){cos}\lambda\right\}\:\Rightarrow \\ $$$${f}\left(\lambda\right)\:=\frac{\pi}{\lambda}{e}^{\left(\mathrm{1}−\lambda\right){sin}\lambda} \:{sin}\left\{\left(\mathrm{1}+\lambda\right){cos}\lambda\right\}\:\:\:\:\:\left(\lambda>\mathrm{0}\right) \\ $$