$\displaystyle \int \frac {x^3 + 13} {x^2 + 5x + 6} dx$
The first step is to do long division. Can anyone tell me if $\displaystyle (x+5) + \frac {-19x - 17} {x^2 + 5x + 6}$ is the right answer?
Hi Archduke,
you can check it quickly...
$\displaystyle x^3+13=\left(x^2+5x+6\right)(x+a)+Rem$
$\displaystyle =x^3+5x^2+6x+ax^2+5ax+6a+Rem$
$\displaystyle =x^3+(5+a)x^2+(6+5a)x+6a+Rem$
Therefore $\displaystyle a=-5$ as $\displaystyle x^3+13$ has no $\displaystyle x^2$ term.
It also has no x term, so we cancel it in the remainder
$\displaystyle x^3+13=x^3-19x-30+19x+43$
$\displaystyle \frac{x^3+13}{x^2+5x+6}=(x-5)+\frac{19x+43}{x^2+5x+6}$