# MAT136H1 Lecture Notes - Partial Fraction Decomposition, Quadratic Function

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.

## 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.