Partial fractions: case when in the denominator a factor occurs with N multiplicity

Question

Suppose we have:

$$\int \frac{3x^3-4x^2-3x+2}{x^2(x+1)(x-1)} ~dx$$

now why do we have to do this with the term with multiplicity greater than $1$?

why the $x^2$ term becomes a sum:

$$\frac{A}{x} +\frac{B}{x^2}+ \frac{C}{(x+1)}+ \frac{D}{(x-1)}$$

Answer 1

When you do partial fraction, you need to check the highest power on the numerator is strictly less than the highest power on the denominator. So for $x^2$ term in the denominator, in general, the numerator has to be at least 1 order less, hence it is linear $ax+b$, we have

$$\frac{ax+b}{x^2}=\frac a x+\frac b{x^2}$$