Question Number 35584 by abdo mathsup 649 cc last updated on 20/May/18
$${let}\:\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{arctan}\left({e}^{−{tx}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} }\:{dx}\:{with}\:{t}>\mathrm{0} \\ $$ $$\left.\mathrm{1}\right)\:{study}\:\:{the}\:{existence}\:{of}\:\:{f}\left({t}\right) \\ $$ $$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({t}\right) \\ $$