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Partial Fraction Decompositions

- Rational Functions
- Partial Fraction Decompositions
- Finding Partial Fractions Decompositions
- Integrating Partial Fraction Decompositions
- Examples

Rational Functions

A function of the type P/Q, where both P and

Q are polynomials, is a rational function.

Definition

Example

The degree of the denominator of the above

rational function is less than the degree of the

numerator. Hence we may rewrite the above

rational function in a simpler form by performing

polynomial division.

Rewriting

For integration, it is always necessary to

perform polynomial division first, if possible.

To integrate the polynomial part is easy, and one

can reduce the problem of integrating a general

rational function to a problem of integrating a

rational function whose denominator has degree

greater than that of the numerator.

Example of Integrating Rational Functions

Here K is the constant of integration.

Partial Fraction Decomposition (1)

of the function to be integrated.

Definition

The above decomposition of the rational function

(3x23x2)/(x3x2x1) is a partial fraction

decomposition.

Partial Fraction Decomposition is a rewriting of

a rational function into a sum a rational

functions with as simple denominators as possible.

General partial fraction decomposition is

technically complicated and involves several

cases. It all starts with a factorization of the

denominator. The type of the partial factor

decomposition depends on the type of the factors

of the denominator. The different cases will be

explained on the followings.

Partial Fraction Decompositions (2)

The partial fraction decomposition of a rational

function RP/Q, with deg(P) lt deg(Q), depends

on the factors of the denominator Q. Since we

are factoring over real numbers, the denominator

Q may have following types of factors

- Simple, non-repeated linear factors ax b.
- Repeated linear factors of the form (ax b)k, k

gt 1. - Simple, non-repeated quadratic factors of the

type ax2 bx c. Since we assume that these

factors cannot anymore be factorized, we have

b2 4 ac lt 0. - Repeated quadratic factors (ax2 bx c)k, kgt1.

Also in this case we have b2 4 ac lt 0.

Finding the partial fraction decompositions is,

in all of the above cases, same type of

computation. Eventually one wants to integrate

the resulting partial fraction decomposition.

The integration can, in all cases, be based on

formulae, but the computations will get

technically complicated.

Simple Linear Factors (1)

Case I

Partial Fraction Decomposition Case I

Simple Linear Factors (2)

Example

Simple Quadratic Factors (1)

Case II

Partial Fraction Decomposition Case II

Simple Quadratic Factors (2)

Example

Repeated Linear Factors (1)

Case III

Partial Fraction Decomposition Case III

Repeated Linear Factors (2)

Example

Repeated Quadratic Factors (1)

Case IV

Partial Fraction Decomposition Case IV

Repeated Quadratic Factors (2)

Example

This can also be computed by Maple command

convert.

Integrating Partial Fraction Decompositions (1)

Partial Fraction Decomposition is the main method

to integrate rational functions. After a general

partial fraction decomposition one has to deal

with integrals of the following types. There are

four cases. Two first cases are easy.

Here K is the constant of integration.

Integrating Partial Fraction Decompositions (2)

In the remaining cases we have to compute

integrals of the type

In the case 3, observe that since the

denominator cannot be factored further we have

4ac-b2 gt 0. By suitable rewritings and

substitutions we get

Here K denotes the constant of integration.

Integrating Partial Fraction Decompositions (3)

By repeated integrations by parts these integrals

can eventually be computed by the formula for the

integrals of type 3.

Theorem

Integration Algorithm

- Integration of a rational function f P/Q,

where P and Q are polynomials can be

performed as follows. - If deg(Q) ? deg(P), perform polynomial division

and write P/Q S R/Q, where S and R are

polynomials with deg(R) lt deg(Q).Integrate the

polynomial S. - Factorize the polynomials Q and R. Cancel the

common factors. Perform Partial Fraction

Decomposition to the simplified form of R/Q. - Integrate the Partial Fraction Decomposition.

Examples (1)

Example 1

Examples (2)

Example 1 (contd)

Substitute ux2x1 in the first remaining

integral and rewrite the last integral.

Examples (3)

Example 2

We can simplify the function to be integrated by

performing polynomial division first. This needs

to be done whenever possible. We get

Partial fraction decomposition for the remaining

rational expression leads to

Now we can integrate

Examples (4)

Example 3

Partial fraction decomposition in this case yields

This is easy to integrate.

It is, however, easier to observe that the

substitution

Leads to the result immediately

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