Note 6.3 - Partial Fractions

Note 6.3 - Partial Fractions

1 Introduction

In this note, we deal with integrations of rational functions. These are functions that almost never have obvious antiderivatives except those like _a+bx¹ :

or _1+x¹2:

or some of their very simple variations. Our goal is to turn a general rational function ^p(x)_q(x) into a combination of these as much as we can.

2 Algebraic Background

Let us review (or preview) some algebraic facts.

Definition 2.1. A real polynomial p(x) is called reducible if p(x) = q(x)r(x) where deg(q) < deg(p). It is called irreducible if it is not reducible.

Turning p into qr above is called a factorization of p. Polynomials, like integers, can be uniquely (almost) factorized:

Theorem 2.2 (Unique Factorization). For every real polynomial p(x), there exist irreducible polynomials q1(x), . . . , qn(x) with deg(qi) ≤ deg(p) and integers a1, . . . , an so that

p(x) = q1(x)^a1· · · qn(x)^an. The above factorization is unique up to a constant multiple.

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Now we turn to rational function ^p(x)_q(x). We may assume that deg(p) < deg(q) (otherwise apply division algorithm). A theorem in algebra says that it has partial fraction decomposition:

Theorem 2.3. For ^p(x)_q(x) above, and q(x) = q₁^a1(x) · · · q_n^an(x) be the factorization of q, the rational function can be written into

p(x)

q(x) =r¹₁(x)

q₁(x)+r¹₂(x)

q²₁(x)+ . . . + r¹_a1

q₁^a1(x)+ . . . + r₁ⁿ(x)

q_n(x)+rⁿ₂(x)

q²_n(x)+ . . . + rⁿ_an q^anⁿ(x), where deg(ri) < deg(qi) for each i.

Of course, this theorem is only practical for low degree polynomials since we have to factor it. Let’s study some example.

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3 Instruction Manual

The instructions are obvious. Rewrite ^p(x)_q(x) into its partial fraction decomposi- tion and see if we can integrate that.

4 Examples

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