Integration of Rational Functions

Wednesday, 8 August 2018

1:33 PM

A rational function  is of the form

Where  and  are polynomials.

 

A polynomial is a function of the form

Where  is a non-negative integer and  are real numbers

 

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A rational fraction  is proper if the degree of  is less than the degree of

 

 is a proper rational function

 is NOT a proper rational function

 

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Method: Long Division to and remainder as a fraction

Then integrate both sides separately

 

Partial Functions

It can be shown using algebra that every proper rational function  can be written as a unique sum of functions of the form

Where the quadratic  is irreducible ()

This sum is called the

 

Any rational function can be integrated with systematic reduction with partial fractions

 

 

How the partial fraction process is carried out depends on the factorisation of the denominator of

 

Distinct Linear Factors

 

Repeated Linear Factors

 

Distinct Quadratic Factors