Question

Помогите решить интеграл, нужно подробное решение. dx/cos^4(5x-1)

Answer 1

 $\int \frac{dx}{cos^4(5x-1)}=\int \frac{1}{cos^2(5x-1)}\cdot \frac{dx}{cos^2(5x-1)} =\\\\=[\, d(tg(5x-1))=(tg(5x-1))'dx=\frac{5}{cos^2(5x-1)}dx\, ]=\\\\=\frac{1}{5}\int (1+tg^2(5x-1))\cdot d(tg(5x-1))=[\, \int (1+t^2)dt=t+\frac{t^3}{3}+C\, ]=\\\\=\frac{1}{5}\cdot (tg(5x-1)+\frac{tg^3(5x-1)}{3})+C$