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Techniques of Integration · Partial Fractions

Partial Fractions: Distinct Linear

P(x)(x−a)(x−b)=Ax−a+Bx−b\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}

Decompose a rational function with distinct linear factors in the denominator into simpler fractions that can be integrated individually.

Conditions. Degree of P(x) must be less than the degree of the denominator.

Worked examples

Find ∫ 1/((x-1)(x+2)) dx.
  1. 1/((x-1)(x+2)) = A/(x-1) + B/(x+2)
  2. Multiply through: 1 = A(x+2) + B(x-1)
  3. x = 1: 1 = 3A → A = 1/3. x = -2: 1 = -3B → B = -1/3
  4. ∫ [(1/3)/(x-1) - (1/3)/(x+2)] dx = (1/3)ln|x-1| - (1/3)ln|x+2| + C

Answer: (1/3)ln|(x-1)/(x+2)| + C

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