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Techniques of Integration · Partial Fractions

Partial Fractions: Repeated Linear

P(x)(x−a)n=A1x−a+A2(x−a)2+⋯+An(x−a)n\frac{P(x)}{(x-a)^n} = \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}

For repeated linear factors, include a term for each power of the factor up to its multiplicity.

Worked examples

Find ∫ (2x+3)/(x+1)² dx.
  1. (2x+3)/(x+1)² = A/(x+1) + B/(x+1)²
  2. 2x+3 = A(x+1) + B. x = -1: 1 = B. Comparing x coefficients: 2 = A.
  3. ∫ [2/(x+1) + 1/(x+1)²] dx = 2ln|x+1| - 1/(x+1) + C

Answer: 2 ln|x+1| - 1/(x+1) + C

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