Question

Помогите пожалуйста:/

Answer 1

$\int \frac{2x^3-1}{x^3+3x^2}\, dx=\int \Big (2-\frac{6x^2+1}{x^2(x+3)}\Big )\, dx=Q\\\\\\\frac{6x^2+1}{x^2(x+3)}=\frac{A}{x^2}+\frac{B}{x}+\frac{C}{x+3}\\\\6x^2+1=A(x+3)+Bx(x+3)+Cx^2\\\\x=0:\; \; A=\frac{6x^2+1}{x+3}=\frac{1}{3}\\\\x=-3:\; \; C=\frac{6x^2+1}{x^2}=\frac{55}{9}\\\\x^2\, |\; 6=B+C\; \; ,\; \; B=6-C=6-\frac{55}{9}=-\frac{1}{9}\\\\\\Q=\frac{1}{3} \int \frac{dx}{x^2}-\frac{1}{9}\int \frac{dx}{x}+\frac{55}{9}\int \frac{dx}{x+3}=-\frac{1}{3x}-\frac{1}{9}\, ln|x|+\frac{55}{9}\, ln|x+3|+C$