Partial fractions

When the numerator degree is at least the denominator degree, divide first. For distinct linear factors, place a constant numerator above each factor. Repeated factors require every power up to the repeated power; an irreducible quadratic generally requires a linear numerator.

Multiply by the common denominator and equate coefficients or substitute convenient values to determine constants. This gives an identity on the domain where the original fraction is defined. Verify by recombining the fractions; integration still requires attention to singularities.

Worked example

Decompose 5/((x + 1)(x + 2)).

1. Write A/(x + 1) + B/(x + 2).
2. Then 5 = A(x + 2) + B(x + 1).
3. At x = −1, A = 5; at x = −2, B = −5.

Answer: 5/(x + 1) − 5/(x + 2), x ≠ −1, −2

Revise first: Algebraic fractions.