Partial Fractions Cheat Sheet, Cheat Sheet of Mathematics

Download Partial Fractions Cheat Sheet and more Cheat Sheet Mathematics in PDF only on Docsity! 1 Harold’s Partial Fractions Cheat Sheet April 2016 Partial Fractions (http://en.wikipedia.org/wiki/Partial_fraction_deco mposition) Condition 𝑓(𝑥) = 𝑃(𝑥) 𝑄(𝑥) = 𝑎𝑥𝑛 + ⋯ + 𝑏 𝑐𝑥𝑚 + ⋯ + 𝑑 where 𝑃(𝑥) 𝑎𝑛𝑑 𝑄(𝑥) are polynomials Preparation Case 1: 𝑛 ≥ 𝑚, Peform long division first Case 2: 𝑛 < 𝑚, Proceed to the cases below Case I: Simple linear (𝟏𝒔𝒕 degree) 𝐴 (𝑎𝑥 + 𝑏) Case II: Multiple degree linear (𝟏𝒔𝒕 degree) 𝐴 (𝑎𝑥 + 𝑏) + 𝐵 (𝑎𝑥 + 𝑏)2 + 𝐶 (𝑎𝑥 + 𝑏)3 Case III: Simple quadratic (𝟐𝒏𝒅 degree) 𝐴𝑥 + 𝐵 (𝑎𝑥2 + 𝑏𝑥 + 𝑐) Case IV: Multiple degree quadratic (𝟐𝒏𝒅 degree) 𝐴𝑥 + 𝐵 (𝑎𝑥2 + 𝑏𝑥 + 𝑐) + 𝐶𝑥 + 𝐷 (𝑎𝑥2 + 𝑏𝑥 + 𝑐)2 + 𝐸𝑥 + 𝐹 (𝑎𝑥2 + 𝑏𝑥 + 𝑐)3 Example Expansion 𝑃(𝑥) (𝑎𝑥 + 𝑏)(𝑐𝑥 + 𝑑)2(𝑒𝑥2 + 𝑓𝑥 + 𝑔) = 𝐴 (𝑎𝑥 + 𝑏) + 𝐵 (𝑐𝑥 + 𝑑) + 𝐶 (𝑐𝑥 + 𝑑)2 + 𝐷𝑥 + 𝐸 (𝑒𝑥2 + 𝑓𝑥 + 𝑔) Typical Solution for Cases I & II ∫ 𝑎 𝑥 + 𝑏 𝑑𝑥 = 𝑎 ln|𝑥 + 𝑏| + 𝐶 Typical Solution for Cases III & IV ∫ 𝑎 𝑥2 + 𝑏2 𝑑𝑥 = 𝑎 𝑏 𝑡𝑎𝑛−1 ( 𝑥 𝑏 ) + 𝐶 Steps to Solve Example 1. Write down problem ∫ 5𝑥 + 1 2𝑥2 − 𝑥 − 1 𝑑𝑥 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 5𝑥 + 1 (2𝑥 + 1)(𝑥 − 1) 4. Expand function with A, B, Cs 5𝑥 + 1 (2𝑥 + 1)(𝑥 − 1) = 𝐴 (2𝑥 + 1) + 𝐵 (𝑥 − 1) 5. Get a common denominator = 𝐴(𝑥 − 1) (2𝑥 + 1)(𝑥 − 1) + 𝐵(2𝑥 + 1) (2𝑥 + 1)(𝑥 − 1) 6. Focus on numerator 5𝑥 + 1 = 𝐴(𝑥 − 1) + 𝐵(2𝑥 + 1) 7. Expand/distribute/FOIL 5𝑥 + 1 = 𝐴𝑥 − 𝐴 + 2𝐵𝑥 + 𝐵