Question

Решить интеграл на фото

Answer 1

$\int \frac{x^3+1}{x^2-x}\, dx=\int (x+1+\frac{x+1}{x^2-x})dx=\int (x+1+\frac{x+1}{x(x-1)})\, dx=\\\\=\int (x+1)\, dx-\int \frac{dx}{x}+\int \frac{2}{x-1}=\frac{(x+1)^2}{2}-ln|x|+2ln|x-1|+C=\\\\=\frac{(x+1)^2}{2}+ln\frac{(x-1)^2}{|x|}+C\\\\\star \; \; \frac{x+1}{x(x-1)}=\frac{A}{x}+\frac{B}{x-1}\\\\x=0:\; A=\frac{1}{-1}=-1\\\\x=1:\; B=\frac{1+1}{1}=2$