Phương trình hoành độ giao điểm $y=\frac{x+1}{x-2}=0\Rightarrow x=-1$

$S=\int\limits_{-1}^{0}{\left| \frac{x+1}{x-2} \right|dx}=\left| \int\limits_{-1}^{0}{\frac{x+1}{x-2}dx} \right|=\left| \int\limits_{-1}^{0}{\left( 1+\frac{3}{x-2} \right)dx} \right|=\left| \left. \left( x+3\ln \left| x-2 \right| \right) \right|_{-1}^{0} \right|=\left| 1+3\ln \frac{2}{3} \right|=3\ln \frac{3}{2}-1$