Question

найти интеграл sin^4(x)/cos^2(x)

Answer 1

$int frac{sin^4x}{cos^2x}, dx=int frac{sin^4xcdot cos^4x}{cos^4xcdot cos^2x}, dx=int frac{sin^4x}{cos^4x}cdot cos^4xcdot frac{dx}{cos^2x}=\\=[1+tg^2x=cos^2x; to ; ; cos^4x=(cos^2x)^2=(1+tg^2x)^2;,\\d(tgx)=frac{dx}{cos^2x}, ]=int tg^4xcdot (1+tg^2x)^2cdot d(tgx)=[, t=tgx, ]=\\=int t^4cdot (1+2t^2+t^4)cdot dt=int (t^4+2t^6+t^8)dt=\\=frac{t^5}{5}+frac{2t^7}{7}+frac{t^9}{9}+C=frac{tg^5x}{5}+frac{2tg^7x}{7}+frac{tg^9x}{9}+C$