Question

Решите неопределённый интеграл

Answer 1

$\displaystyle \int\frac{dx}{cos^4x-sin^4x}=\int\frac{\displaystyle\frac{dx}{cos^4x}}{1-tg^4x}=\int\frac{1+tg^2x}{1-tg^4x}d(tgx)=\\=\int\frac{d(tgx)}{1-tg^2x}=\frac{1}{2}ln|\frac{1+tgx}{1-tgx}|+C$

$\displaystyle\int\frac{dx}{\sqrt x+\sqrt[4]x}=4\int\frac{t^2dt}{t+1}=4\int(t-1+\frac{1}{t+1})dt=\\=4(\frac{t^2}{2}-t+ln|t+1|)+C=2\sqrt x-4\sqrt[4]x+4ln|\sqrt[4]x+1|+C\\\\x=t^4;dx=4t^3dt$

$\displaystyle \int\frac{xdx}{cos^2x}=xtgx-\int tgxdx=xtgx+ln|cosx|+C\\u=x;du=dx\\dv=\frac{dx}{cos^2x};v=tgx$