Question

Помогите решить, найти неопределенный интеграл (cos2x)/cos^2xsin^2x dx

Answer 1

$int frac{cos2x}{cos^2 x*sin^2x}dx = int frac{cos^2x-sin^2x}{cos^2 x*sin^2x}dx = int frac{cos^2x}{cos^2 x*sin^2x}dx -int frac{sin^2x}{cos^2 x*sin^2x}dx =\\ int frac{dx}{sin^2x} -int frac{dx}{cos^2 x}=-ctgx-tgx+C=-( frac{cosx}{sinx} + frac{sinx}{cosx} )+C=\\ =-( frac{cos^2x+sin^2x}{sinxcosx})+C=-frac{1}{sinxcosx}+C$