Question

ИНТЕГРАЛ ОТ: $frac{2 x^{2} -5x+1 }{ x^{3} -2 x^{2} +1} dx$

Answer 1

$frac{2x^2-5x+1}{x^3-2x^2+1} = frac{2x^2-5x+1}{(x-1)(x^2-x-1)}=frac{2x^2-5x+1}{(x-1)(x-frac{1-sqrt5}{2})(x- frac{1+sqrt5}{2} )} =\\= frac{A}{x-1}+frac{B}{x-frac{1-sqrt5}{2}}+frac{C}{x-frac{1+sqrt5}{2}} ; ;\\2x^2-5x+1=A(x-frac{1-sqrt5}{2})(x-frac{1+sqrt5}{2}}{})+B(x-1)(x-frac{1+sqrt5}{2})+\\+C(x-1)(x- frac{1-sqrt5}{2} ); ;$

$x=1:; ; A= frac{-2}{(1- frac{1-sqrt5}{2})(1-frac{1+sqrt5}{2})}=frac{-2}{-1} =2\\x= frac{1-sqrt5}{2} ;; ; B= frac{(3+3sqrt5)/2}{(5+sqrt5)/2}=frac{3}{sqrt5}$

$x= frac{1+sqrt5}{2} :; ; C= frac{(3-3sqrt5)/2}{sqrt5(sqrt5-1)/2} =-frac{3}{sqrt5}; ;\\\int frac{2x^2-5x+1}{x^3-2x^2+1} dx=2int frac{dx}{x-1}+frac{3}{sqrt5}int frac{dx}{x-frac{1-sqrt5}{2}}-frac{3}{sqrt5}int frac{dx}{x-frac{1+sqrt5}{2}} =\\=2ln|x-1|+frac{3}{sqrt5}cdot lnleft |x-frac{1-sqrt5}{2}right |-frac{3}{sqrt5}cdot lnleft |x-frac{1+sqrt5}{2}right |+C$