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Partial Fractions Decomposition
AP Calculus BC · Unit 6 Integration
Learning Objectives
Decompose P(x)/Q(x) into sums of simpler rational parts
Solve for coefficients using Algebraic or Heaviside method
Integrate using natural log: ∫1/(ax+b) dx = (1/a)ln|ax+b|
Heaviside Cover-Up
For distinct linear factors: A / (x−a)
To find A:
1. "Cover up" (x−a) in denominator
2. Plug x = a into the rest of the expression
3. The result is A. Fast and robust!
Tags
integration
partial fractions
heaviside
natural log
INTEGRAL ∫ P(x)/Q(x) dx
∫ (5x−3)/(x²−2x−3) dx
∫ 1/(x²−a²) dx (Generic Form)
∫ (x−1)/(x³+3x²+2x) dx
∫ (2x+1)/(x²+x−2) dx
Degree Rule
If Degree(Numerator) ≥ Degree(Denominator), you MUST use polynomial long division BEFORE partial fractions.