Question Number 28886 by abdo imad last updated on 31/Jan/18
$${find}\:\int\:\:\:\frac{{x}}{{cos}^{\mathrm{2}} {x}}{dx}. \\ $$
Answered by mrW2 last updated on 01/Feb/18
$$\:\int\:\:\:\frac{{x}}{{cos}^{\mathrm{2}} {x}}{dx} \\ $$$$=\int{x}\:{d}\left(\mathrm{tan}\:{x}\right) \\ $$$$={x}\:\mathrm{tan}\:{x}−\int\mathrm{tan}\:{x}\:{dx} \\ $$$$={x}\:\mathrm{tan}\:{x}+\int\frac{\mathrm{1}}{\mathrm{cos}\:{x}}\:{d}\left(\mathrm{cos}\:{x}\right) \\ $$$$={x}\:\mathrm{tan}\:{x}+\mathrm{ln}\:\left(\mathrm{cos}\:{x}\right)+{C} \\ $$