Question

Решите интеграл: dx/(sin^2 (x)*(1+cos (x))

Answer 1

$int frac{dx}{sin^2x(1+cosx)} =[; t=tg frac{x}{2}; ,; sinx= frac{2t}{1+t^2}; ,; cosx= frac{1-t^2}{1+t^2}; ,\\dx= frac{2, dt}{1+t^2} ; ]=int frac{2, dt}{(1+t^2)cdot frac{4t^2}{(1+t^2)^2}cdot (1+frac{1-t^2}{1+t^2} )} =int frac{2, dt}{frac{4t^2}{1+t^2}cdot frac{1+t^2+1-t^2}{1+t^2} } =\\=int frac{(1+t^2)^2, dt}{2t^2cdot 2} = frac{1}{4}cdot int frac{1+2t^2+t^4}{t^2} dt= frac{1}{4}cdot int (t^{-2}+2+t^2)dt=$

$= frac{1}{4}cdot Big (frac{t^{-1}}{-1}+2t+frac{t^3}{3})= frac{1}{4}cdot (-frac{1}{tgfrac{x}{2}}+2, tg frac{x}{2} + frac{1}{3}, tg^3, frac{x}{2})+C$