Question

Помогите,пожалуйста!!
1. Integral x^2/x^6-1 dx
2. Integral 1/ cos^2x (3-2tg x) dx
3. Integral sin 2x/1+ cos^2x dx

Answer 1

$1); ; int frac{x^2}{x^6-1}dx=frac{1}{3}int frac{3x^2, dx}{(x^3)^2-1} =[; t=x^3; ,; dt=3x^2, dx; ]=\\= frac{1}{3}cdot int frac{dt}{t^2-1} = frac{1}{3} cdot frac{1}{2} cdot lnBig | frac{t-1}{t+1} Big |+C= frac{1}{6} cdot lnBig | frac{x^3-1}{x^3+1} Big |+C$

$2); ; int frac{dx}{cos^2x(3-2tgx)} =[; t=3-2tgx; ,; dt=frac{-2, dx}{cos^2x}; ]=-frac{1}{2}int frac{dt}{t} =\\=-frac{1}{2}cdot ln|t|+C=- frac{1}{2} cdot ln|3-2tgx|+C\\3); ; int frac{sin2x, dx}{1+cos^2x} =[; t=1+cos^2x; ,; dt=-2cosx, sinx, dx=-sin2x, dx; ]=\\=-int frac{dt}{t} =-ln|t|+C=-ln|1+cos^2x|+C$