Question Number 33644 by mondodotto@gmail.com last updated on 21/Apr/18
$$\:\int\left(\boldsymbol{\mathrm{sec}}^{\mathrm{2}} \boldsymbol{{x}}\right)\boldsymbol{{e}}^{\boldsymbol{\mathrm{tan}{x}}} \boldsymbol{{dx}} \\ $$
Answered by Joel578 last updated on 21/Apr/18
$${I}\:=\:\int\:\left(\mathrm{sec}^{\mathrm{2}} \:{x}\right)\:{e}^{\mathrm{tan}\:{x}} \:{dx} \\ $$$${u}\:=\:\mathrm{tan}\:{x}\:\:\rightarrow\:\:{du}\:=\:\mathrm{sec}^{\mathrm{2}} \:{x}\:{dx} \\ $$$$ \\ $$$${I}\:=\:\int\:{e}^{{u}} \:{du} \\ $$$$\:\:\:=\:{e}^{{u}} \:+\:{C}\:\:\:=\:{e}^{\mathrm{tan}\:{x}} \:+\:{C} \\ $$