Question

Решите неопределённый интеграл

Answer 1

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

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

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