(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[tex]\int[/tex](3x^{3}-4x^{2}-3x+2)/(x^{4}-x^{2})

2. Relevant equations

P(x)/Q(x)=A_{1}/(x-r_{1})+A_{2}/(x-r_{2})+...

if x-r occurs with multiplicity m, then A/(x-r) must be replaced by a sum of the form:

B_{1}/(x-r)+B_{2}/(x-r)^{2}+...

I think this second equation is the source of my confusion.

3. The attempt at a solution

I began by factoring the denominator:

x^{4}-x^{2}= x2(x+1)(x-1)

So, according to my book, we have the following constants:

A/x + B/x^{2}+ C/(x+1) + D/(x-1)

First question: where did the x come from in the constant A/x? Does this follow from the rule above? That is, because x^{2}has multiplicity 2, I get A/x and B/x^{2}?

Next, according to my book, when you clear the fractions, you should get:

Ax(x+1)(x-1) + B(x+1)(x-1) + Cx^{2}(x-1) + Dx^{2}(x+1)

I don't understand this. Why isn't it Ax^{2}(x+1)(x-1) + Bx(x+1)(x-1) + Cx^{3}(x-1) + Dx^{3}(x+1)?

Can someone explain? Many thanks.

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# Homework Help: Integration using partial fractions

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