Integration of simple rational fractions. Integration of rational fractions
LECTURE PLAN:
1. Selecting the proper rational fraction
2. Integrating Proper Rational Functions
3. Integrating Improper Rational Functions
Selecting the proper rational fraction
Suppose is a rational function; that is, and are polynomial functions. If the degree of is greater than or equal to the degree of , then by long division, where is a proper rational fraction; that is, the degree of is less than the degree of . A theorem in advanced algebra states that every proper rational function can be expressed as a sum
where are rational functions of the form
or
in which the denominators are factors of . The sum is called the partial fraction decomposition of . The first step is finding the form of the partial fraction decomposition of is to factor completely into linear and irreducible quadratic factors, and then collect all repeated factors so that is expressed as a product of distinct factors of the form
and .
From these factors we can determine the form of the partial fraction decomposition using the following two rules:
Linear Factor Rule: For each factor of the form , the partial fraction decomposition contains the following sum of m partial fractions:
where A_{1}, A_{2}, . . ., A_{m} are constants to be determined.
Quadratic Factor Rule: For each factor of the form , the partial fraction decomposition contains the following sum of m partial fractions:
where A_{1}, A_{2}, . . ., A_{m}, B_{1}, B_{2}, …, B_{m} are constants to be determined.
I. Integrating Proper Rational Functions
Example 10: Find .
using the Linear Factor Rule, we get
after
multiplying by . If we let , then ; if we let
, then . Thus, =
.
Example 11: Find .
by the Linear Factor Rule, we get
after multiplying
by . If we let x = 2, then ; if we let x = 3, then
. Thus,
.
II. Integrating Improper Rational Functions
Although the method of partial fractions only applies to proper rational functions, an improper rational function can be integrated by performing long division (or synthetic division). If is a rational function where and are polynomial functions and the degree of is greater than or equal to the degree of , then by long division, where is a proper rational fraction. Since is a proper rational function, it can be decomposed into partial fractions.
Example 12: Find .
By synthetic division, 1 1 0 0 1 .
1 1 1
1 1 1 2
Thus,
.
Date: 20150102; view: 1112
