Question

$\int\limit \frac{ x^{2} -x-2}{x-2} dx$

Answer 1

$\int\limits { \frac{ x^{2} -x-2}{x-2} } \, dx = \int\limits { \frac{ x^{2} -2x+x-2}{x-2} } \, dx = \int\limits{ \frac{ x^{2} -2x}{x-2} } \, dx + \int\limits { \frac{x-2}{x-2} } \, dx$
$= \int\limits {x} \, dx + \int\limits\, dx = \frac{ x^{2} }{2} +x$

Answer 2

$x^2-x-2=(x+1)(x-2)$.
Deci
 $\displaystyle\int\frac{x^2-x-2}{x-2}dx=\int\frac{(x+1)(x-2)}{x-2}dx=\\=\displaystyle\int (x+1)dx=\frac{x^2}{2}+x+C$