Partial Fractions

You’ll be able to

Decompose a rational function into a sum of simpler fractions
Solve for the unknown constants in a decomposition
Integrate the resulting terms, typically to logarithms

Splitting a rational function

A partial-fraction decomposition rewrites a rational function whose denominator factors into a sum of simpler fractions, each with one factor of the denominator. This is the reverse of finding a common denominator, and it makes integration easy: each simple piece integrates to a logarithm or an arctangent. The method requires that the numerator degree be less than the denominator degree; if not, you do polynomial long division first.

Decomposition over distinct linear factors

P(x) / [(x − a)(x − b)] = A/(x − a) + B/(x − b)

One unknown constant per distinct linear factor. Solve for A and B by clearing denominators and substituting convenient x-values.

Finding the constants and integrating

To find the constants, multiply both sides by the full denominator to clear fractions, then either substitute the roots (each choice of x zeroes out all but one unknown) or equate coefficients. Once decomposed, each term ∫A/(x − a) dx integrates to A·ln|x − a| + C. The result is almost always a sum of natural logs, which is why partial fractions is the standard route for rational integrands the basic rules cannot touch.

Worked example

Evaluate ∫ 1 / [x(x − 1)] dx using partial fractions.

1.Set up the decomposition: 1/[x(x − 1)] = A/x + B/(x − 1).
2.Clear denominators: 1 = A(x − 1) + Bx. Substitute x = 0: 1 = A(−1), so A = −1. Substitute x = 1: 1 = B(1), so B = 1.
3.Rewrite the integral: ∫[−1/x + 1/(x − 1)] dx.
4.Integrate each term: −ln|x| + ln|x − 1| + C.

Answer: ∫ 1/[x(x − 1)] dx = ln|x − 1| − ln|x| + C = ln|(x − 1)/x| + C.

Watch out

Partial fractions only applies when the numerator degree is strictly less than the denominator degree. If the fraction is improper (top degree ≥ bottom degree), you must do polynomial long division first, then decompose the remainder.

Checkpoint

What is the correct partial-fraction form for 5 / [(x − 2)(x + 3)]?

Checkpoint

After decomposing, an integral becomes ∫ [2/(x − 1)] dx. What is the result?

On the exam

The fastest way to find the constants is the "cover-up" method: to get the constant over (x − a), cover that factor and substitute x = a into what remains. It computes each numerator in one step, ideal under time pressure.

Answer the 2 checkpoints as you read.