Question

Решите интеграл (x+2/x^2(x-1)) *dx

Answer 1

$intlimits^3_2 frac{x+2}{x^2(x-1)} , dx =Q\\ frac{x+2}{x^2(x-1)}=frac{A}{x^2}+frac{B}{x}+frac{C}{x-1} \\x+2=A(x-1)+Bx(x-1)+Cx^2\\x=0:; ; A= frac{2}{-1}=-2\\x=1:; ; C= frac{1+2}{1}=3\\x^2|; 0=B+C; ' ,; ; B=-C=-3\\intlimits frac{x+2}{x^2(x-1)}, dx=-2, intlimits frac{dx}{x^2}-3, intlimits frac{dx}{x} +3, intlimits frac{dx}{x-1}=\\=-2cdot frac{-1}{x}-3, ln|x|+3, ln|x-1|+C=frac{2}{x}+3, ln| frac{x-1}{x} |+C$

$Q= (frac{2}{x}+2, ln| frac{x-1}{x}|)Big |_2^3= frac{2}{3}+2, ln frac{2}{3}-1-2, ln frac{1}{2}=- frac{1}{3}+2, ln frac{4}{3}$