Partial Fractions for Rational Integrands
Split a rational function into simpler fractions whose integrals are logarithms and arctangents, after checking the degree of the numerator first.
What a learner can do afterwards
- Divide first when the numerator has degree at least that of the denominator
- Set up the correct form for a repeated linear factor
- Integrate the resulting pieces and combine the logarithms
1 · Read
Partial fractions only start on a proper fraction, where the top degree is below the bottom degree. When the top is heavy, divide first: (x^2 + 3x + 5)/(x + 1) becomes x + 2 plus 3/(x + 1). Only the proper remainder gets split.
Take 1/(x^2 - 1). Factor the bottom into (x - 1)(x + 1) and write A/(x - 1) + B/(x + 1). Strategic substitution finds them: x = 1 gives A = 1/2, and x = -1 gives B = -1/2. Each piece then integrates to a log: (1/2) ln|x - 1| - (1/2) ln|x + 1| + C.
Repeated factors need one term per power. For 5/(x + 2)^2 write A/(x + 2) + B/(x + 2)^2, not the top power alone. The same holds for x^2(x + 3): use A/x + B/x^2 + C/(x + 3).
Each split piece integrates to a logarithm, or an arctan for a quadratic piece. Gather the logs at the end into one compact answer. Always check by recombining over a common denominator.
Divide first when the top is heavy, then split and gather logarithms.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.