Question

$intlimits frac{dx}{3+5cosx}$

Answer 1

Воспользуемся универсальной тригонометрической подстановкой.
$displaystyle...=bigg{u=tgbigg( frac{x}{2} bigg);,,sin x= frac{2u}{u^2+1} ;,, cos x= frac{1-u^2}{u^2+1} ;,,,dx= frac{2du}{u^2+1}bigg}=\ \ \ =intlimits { frac{2du}{(u^2+1)( frac{5(1-u^2)}{1+u^2} +3)} } , =intlimits { frac{du}{4-u^2} } , = frac{1}{4} lnbigg| frac{2+u}{2-u} bigg|+C =\ \ \ =boxed{ frac{1}{4} lnbigg| frac{2+ tg frac{x}{2} }{2-tg frac{x}{2} } bigg|+C}$