1
Haroldโs Partial Fractions
Cheat Sheet
April 2016
Partial Fractions
(http://en.wikipedia.org/wiki/Partial_fraction_deco
mposition)
Condition
๐(๐ฅ)=๐(๐ฅ)
๐(๐ฅ)=๐๐ฅ๐+ โฏ+๐
๐๐ฅ๐+ โฏ+๐
where ๐(๐ฅ) ๐๐๐ ๐(๐ฅ) are polynomials
Preparation
Case 1: ๐ โฅ ๐, Peform long division first
Case 2: ๐ < ๐, Proceed to the cases below
Case I: Simple linear (๐๐๐ degree)
๐ด
(๐๐ฅ+ ๐)
Case II: Multiple degree linear (๐๐๐ degree)
๐ด
(๐๐ฅ+ ๐)+ ๐ต
(๐๐ฅ+ ๐)2+ ๐ถ
(๐๐ฅ+ ๐)3
Case III: Simple quadratic (๐๐๐
degree)
๐ด๐ฅ+ ๐ต
(๐๐ฅ2+๐๐ฅ +๐)
Case IV: Multiple degree quadratic (๐๐๐
degree)
๐ด๐ฅ+ ๐ต
(๐๐ฅ2+๐๐ฅ +๐)+๐ถ๐ฅ+ ๐ท
(๐๐ฅ2+๐๐ฅ +๐)2+๐ธ๐ฅ+๐น
(๐๐ฅ2+๐๐ฅ +๐)3
Example Expansion
๐(๐ฅ)
(๐๐ฅ+ ๐)(๐๐ฅ +๐)2(๐๐ฅ2+๐๐ฅ+ ๐)
=๐ด
(๐๐ฅ+ ๐)+๐ต
(๐๐ฅ+ ๐)+๐ถ
(๐๐ฅ+ ๐)2+๐ท๐ฅ +๐ธ
(๐๐ฅ2+๐๐ฅ +๐)
Typical Solution for Cases I & II
โซ๐
๐ฅ+ ๐ ๐๐ฅ = ๐ ln|๐ฅ+๐|+ ๐ถ
Typical Solution for Cases III & IV
โซ๐
๐ฅ2+๐2 ๐๐ฅ =๐
๐ ๐ก๐๐โ1 (๐ฅ
๐)+๐ถ
Steps to Solve
Example
1. Write down problem
โซ5๐ฅ+ 1
2๐ฅ2โ ๐ฅโ 1 ๐๐ฅ
2. Check if long division is needed
Not needed since degree of numerator (top) is less
than degree of denominator (bottom)
3. Factor denominator of function
5๐ฅ+ 1
(2๐ฅ+ 1)(๐ฅโ 1)
4. Expand function with A, B, Cs
5๐ฅ+ 1
(2๐ฅ + 1)(๐ฅโ 1) =๐ด
(2๐ฅ+ 1)+๐ต
(๐ฅโ 1)
5. Get a common denominator
=๐ด(๐ฅ โ1)
(2๐ฅ+ 1)(๐ฅโ 1)+๐ต(2๐ฅ +1)
(2๐ฅ+ 1)(๐ฅโ 1)
6. Focus on numerator
5๐ฅ + 1 = ๐ด(๐ฅโ1)+ ๐ต(2๐ฅ + 1)
7. Expand/distribute/FOIL
5๐ฅ + 1 = ๐ด๐ฅโ ๐ด + 2๐ต๐ฅ +๐ต
