MAT136H1 Lecture Notes - Partial Fraction Decomposition, Quadratic Function

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7.4 Integration Techniques
Partial Fractions for Rational Functions: Overview
Rational functions are functions with a polynomial in the numerator and the denominator, so that:


If the degree of  is less than , it is called proper rational function.
If the degree of  is greater than , it is called improper rational function. Then before
proceeding to applying partial fractions, it must be written in the form of 
 
,
where  is the remainder obtained from long division.
4 Cases of Rational Functions
Case 1: When the function in the denominator can be factored into linear functions. Then split the
function in the numerator and integrate separately.
Case 2: Similar to Case 1, but some of the factored terms in the denominator have order greater than 1.
Then write out each power occurrence, and decompose the numerator accordingly, and then integrate
separately.
Case 3: When the factored term in the denominator is irreducible quadratic factor, then after
decomposing the numerator and splitting into separate integrals, for those terms with irreducible
quadratic denominator, apply the formula: 


  
Case 4: Similar to Case 3 but like Case 2, some of the irreducible quadratic terms are of power greater
than 1. Then write each occurrence for every power. Then decompose the numerator, split into
separate integrals, and apply the above formula where necessary.
Irreducible quadratic factor means its discriminant is negative, so for any quadratic function 
  , where    . These cannot be factored over real numbers.
Rationalizing Substitutions
Where the integral has a function in the form of 
, then apply substitution   
.
Then can be raised to whatever power necessary to remove the root sign. Then the substituted
relationship can be rearranged with respect to variable , and so the relationship between  and 
can be established, ready to be substituted into the integral for easier evaluation.
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Document Summary

Rational functions are functions with a polynomial in the numerator and the denominator, so that: If the degree of ( ) is less than ( ), it is called proper rational function. If the degree of ( ) is greater than ( ), it is called improper rational function. Then before proceeding to applying partial fractions, it must be written in the form of ( ) where ( ) is the remainder obtained from long division. Case 1: when the function in the denominator can be factored into linear functions. Then split the function in the numerator and integrate separately. Case 2: similar to case 1, but some of the factored terms in the denominator have order greater than 1. Then write out each power occurrence, and decompose the numerator accordingly, and then integrate separately.

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