ИНТЕГРАЛ ОТ:

Решите задачу:

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

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

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