Question

Интеграл dx/ (sin^2(x)*(1-cos(x)))

Answer 1

$large intfrac{dx}{sin^2x(1-cosx)}=intfrac{2(1+t^2)^3}{8(1+t^2)t^4}dt=frac{1}{4}intfrac{(1+t^2)^2}{t^4}dt=\\=frac{1}{4}int(1+frac{2}{t^2}+frac{1}{t^4})dt=frac{t}{4}-frac{1}{2t}-frac{1}{12t^3}+C=\\=frac{tgfrac{x}{2}}{4}-frac{ctgfrac{x}{2}}{2}-frac{ctg^3frac{x}{2}}{12}+C\\\t=tgfrac{x}{2}\x=2arctgt\dx=frac{2dt}{1+t^2}\sinx=frac{2t}{1+t^2} ;cosx=frac{1-t^2}{1+t^2}$