Question Number 59160 by maxmathsup by imad last updated on 05/May/19
$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({xcos}\theta\right)}{{x}^{\mathrm{2}} \:+\theta^{\mathrm{2}} }\:{d}\theta\:\:\:\:\:\:{and}\:\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{sin}\left({xcos}\theta\right)}{{x}^{\mathrm{2}} \:+\theta^{\mathrm{2}} }\:{d}\theta \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right)\:{and}\:{g}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{cos}\theta\right)}{\mathrm{4}+\theta^{\mathrm{2}} }\:{d}\theta\:\:{and}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}\left(\mathrm{2}{cos}\theta\right)}{\mathrm{4}+\theta^{\mathrm{2}} }\:{d}\theta \\ $$$$\left.\mathrm{3}\right)\:{let}\:{u}_{{n}} ={f}\left({n}^{\mathrm{2}} \right)\:\:\:{study}\:\:{the}\:{serie}\:\Sigma\:{u}_{{n}} \\ $$