Question

Вычислить интеграл: ∫(x² + x + 2)dx / [x² * (x - 3)]

Answer 1

$\int \frac{(x^2+x+2)dx}{x^2(x-3)} =I\\\\ \frac{x^2+x+2}{x^2(x-3)}=\frac{A}{x^2}+ \frac{B}{x}+\frac{C}{x-3} = \frac{A(x-3)+Bx(x-3)+Cx^2}{x^2(x-3)} \; ;\\\\x^2+x+2=(B+C)x^2+(A-3B)x-3A\\\\1=B+C\\\\1=A-3B\\\\2=-3A\; \; \to \; \; A=-\frac{2}{3}\\\\1=-\frac{2}{3}-3B\; \; \to \; \; \; B=-\frac{5}{9}\\\\1=-\frac{5}{9}+C\; \; \to \; \; \; C=\frac{14}{9}\\\\\\I=-\frac{2}{3}\cdot \int \frac{dx}{x^2}-\frac{5}{9}\cdot \int \frac{dx}{x} +\frac{14}{9}\cdot \int \frac{dx}{x-3}=$

$=-\frac{2}{3}\cdot (-\frac{1}{x})-\frac{5}{9}\cdot ln|x|+\frac{14}{9}\cdot ln|x-3|+C$