Question Number 12702 by mad last updated on 29/Apr/17
$${find}\:\int{cos}^{\mathrm{2}} \mathrm{2}{x}\:{dx} \\ $$
Answered by Joel577 last updated on 29/Apr/17
$$\mathrm{cos}^{\mathrm{2}} \:\mathrm{2}{x}\:=\:\frac{\mathrm{1}\:+\:\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{2}} \\ $$$$\int\:\mathrm{cos}^{\mathrm{2}} \:\mathrm{2}{x}\:{dx} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\int\:\left(\mathrm{1}\:+\:\mathrm{cos}\:\mathrm{4}{x}\right)\:{dx} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\int\:\mathrm{1}\:{dx}\:+\:\int\:\mathrm{cos}\:\mathrm{4}{x}\:{dx}\right) \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\left({x}\:+\:\frac{\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{4}}\right)\:+\:{C} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}{x}\:+\:\frac{\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{8}}\:+\:{C} \\ $$