Calculus: Integration of Rational functions

A rational function is by definition the quotient of two polynomials. For example

displaymath22

are all rational functions. Remember in the definition of a rational function, you will not see tex2html_wrap_inline24 neither tex2html_wrap_inline26 or |x| for example. Note that integration by parts will not be enough to help integrate a rational function. Therefore, a new technique is needed to do the job. This technique is called Decomposition of rational functions into a sum of partial fractions (in short Partial Fraction Decomposition).
Let us summarize the practical steps how to integrate the rational function tex2html_wrap_inline30 :

1
If tex2html_wrap_inline32 , perform polynomial long-division. Otherwise go to step 2.
2
Factor the denominator Q(x) into irreducible polynomials: linear and irreducible quadratic polynomials.
3
Find the partial fraction decomposition.
4
Integrate the result of step 3.

Remark: The main difficulty encountered in general when using this technique is in dealing with step 2 and step 3. Therefore, it is highly recommended to do a serious review of partial decomposition technique before adventuring into integrating fractional functions.

The following examples illustrate cases in which you will be required to use Partial Fraction Decomposition technique:

[Calculus] [Integration By Parts]
[Geometry] [Algebra] [Trigonometry ]
[Differential Equations] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi

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