3.1.4—Partial fractions

Decompose a proper rational function into simpler fractions whose denominators are the linear or irreducible factors of the original denominator. Repeated factors require repeated terms.

Factor the denominator first, write the correct unknown numerators, clear denominators and equate coefficients. If the fraction is improper, divide before decomposing.

1/[x(x+1)] = 1/x − 1/(x+1), making its integral ln|x|−ln|x+1|+C.

A repeated factor such as (x−1)² needs A/(x−1)+B/(x−1)²; one term cannot represent both powers.