<s>[INST] Finding partial fractions expansions mentally On a problem on a test, my students were asked to find $\displaystyle\int\frac{6x^4-7x^3-13x-6}{x^3-2x^2} dx$,
and one student began by writing $\displaystyle\int\frac{6x^4-7x^3-13x-6}{x^2(x-2)} dx=\int\left(\frac{3}{x^2}+6x+\frac{2}{x-2}+\frac{8}{x}+5\right)dx$.
My question is how someone can get this result without doing any division or partial fraction decomposition -- what techniques can be used to get this?
 [/INST] $$  \frac{6x^4 - 12 x^3 + 5 x^3 - 10 x^2 + 8 x^2 - 16 x + 2 x^2 + 3x-6 }{x^2(x-2)} $$
$$  \frac{(6x^4 - 12 x^3) + (5 x^3 - 10 x^2) + (8 x^2 - 16 x) + 2 x^2 + (3x-6) }{x^2(x-2)} $$
My favorite high school math teacher called the rule a "propitious zero," adding and subtracting the same thing to get a more attractive grouping.
In the order I wrote things,
$$ 6x + 5 + \frac{8}{x} + \frac{2}{x-2} + \frac{3}{x^2} $$
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