Question

найти интеграл sin^4(x)/cos^2(x)

Answer 1

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