Question

ПОМОГИТЕ!НАЙДИТЕ ИНТЕГРАЛ!

Answer 1

$int frac{dx}{5-3cosx}= [, t=tgfrac{x}{2},; cosx= frac{1-t^2}{1+t^2} ,; dx= frac{2dt}{1+t^2}, ]=\\=int frac{ frac{2dt}{1+t^2} }{5-3cdot frac{1-t^2}{1+t^2}} =int frac{2dt}{5(1+t^2)-3(1-t^2)} =int frac{dt}{4t^2+1}=frac{1}{4}int frac{dt}{t^2+(frac{1}{2})^2}=\\=frac{1}{4}cdot frac{1}{frac{1}{2}}}arctgfrac{t}{frac{1}{2}}+C=frac{1}{2}cdot arctg(2tgfrac{x}{2})+C$