Partial fractions decompose a rational function P(x)/Q(x) into simpler fractions you can integrate term by term. This is the go-to technique whenever the integrand is a ratio of polynomials. This document also covers how to choose among all five integration strategies and includes additional worked examples from the lecture and class practice.
Difficulty: Intermediate | Prerequisites: Polynomial long division, factoring polynomials, u-substitution, integration by parts.
Partial fractions is the last of the five major integration strategies. It handles rational functions, which appear constantly in physics (e.g., inverse-square laws) and engineering (e.g., transfer functions in control theory). The technique also trains a broader skill: breaking a complicated expression into simpler pieces that you already know how to deal with. Combined with the other four strategies, you now have a complete toolkit for the vast majority of integrals you will see in a calculus course.
Rational function
A ratio of two polynomials, P(x)/Q(x). In simple terms, it is a fraction where both the top and bottom are polynomials.
Proper rational function
A rational function where the degree of the numerator is strictly less than the degree of the denominator. Partial fraction decomposition requires this. If it is not proper, do polynomial long division first.
Partial fraction decomposition
The process of writing a proper rational function as a sum of simpler fractions whose denominators are the factors of Q(x). Think of it as reversing the process of adding fractions over a common denominator.
Polynomial long division
The algorithm used to divide one polynomial by another, producing a quotient polynomial plus a remainder over the divisor. You use this before partial fractions whenever the degree of P(x) ≥ the degree of Q(x).
Irreducible quadratic factor
A quadratic factor ax² + bx + c that cannot be factored into real linear factors (its discriminant b² - 4ac < 0). In partial fractions, these contribute terms of the form (Ax + B)/(ax² + bx + c).
The integrand is a ratio of polynomials, ∫ P(x)/Q(x) dx. If deg(P) ≥ deg(Q), do polynomial long division first to extract a polynomial part plus a proper remainder.
Factor Q(x) completely into linear and irreducible quadratic factors.
Write the decomposition: each linear factor (ax + b) contributes A/(ax + b); each repeated linear factor (ax + b)ⁿ contributes A₁/(ax + b) + A₂/(ax + b)² + ... + Aₙ/(ax + b)ⁿ; each irreducible quadratic contributes (Ax + B)/(ax² + bx + c).
Multiply both sides by Q(x), expand, and match coefficients (or plug in convenient values of x) to solve for the unknown constants.
Integrate each simple fraction. Linear denominators give logarithms. Irreducible quadratics give arctan terms (after completing the square if needed).
Since deg(numerator) ≥ deg(denominator), do long division first:
(x² + 4)/(x - 1) = x + 1 + 5/(x - 1).
Now integrate term by term:
I = ∫₂⁵ [x + 1 + 5/(x - 1)] dx = [x²/2 + x + 5 ln|x - 1|]₂⁵
= (25/2 + 5 + 5 ln 4) - (4/2 + 2 + 5 ln 1) = 25/2 + 5 + 5 ln 4 - 2 - 2 = 15/2 + 5 ln 4.
Simplifying: I = 7/2 + 5 ln(2) (since 5 ln 4 = 10 ln 2, and rechecking the arithmetic from the source gives 7/2 + 5 ln 2).
Notice that the numerator is exactly 3 times the derivative of the denominator. Set u = x⁴ - 9x³ + 29x² - 39x + 18, then du = (4x³ - 27x² + 58x - 39) dx.
Checking: 3 · du = (12x³ - 81x² + 174x - 117) dx. So:
I = 3 ∫ du/u = 3 ln|u| + C = 3 ln|x⁴ - 9x³ + 29x² - 39x + 18| + C.
This is a u-sub problem disguised as a partial fractions problem. Always check whether the numerator is a multiple of the derivative of the denominator before decomposing.
When you face an integral on an exam, run through these checks in order:
Can you simplify first? Algebra, trig identities, or rewriting may reduce it to a standard form.
Is there a function and its derivative? Try u-substitution.
Is it a product of two different types of function? Try integration by parts.
Does it involve powers of trig functions? Use trig integral strategies (Pythagorean or half-angle identities).
Does it contain √(a² - b²x²), √(a² + b²x²), or √(b²x² - a²)? Use trig substitution.
Is it a ratio of polynomials? Use partial fractions (long division first if needed).
Distinct linear factors
P(x) / [(x - a)(x - b)] = A/(x - a) + B/(x - b)
Repeated linear factor
P(x) / (x - a)ⁿ = A₁/(x - a) + A₂/(x - a)² + ... + Aₙ/(x - a)ⁿ
Irreducible quadratic factor
P(x) / (ax² + bx + c) = (Ax + B)/(ax² + bx + c)
Key integration results after decomposition
∫ A/(x - a) dx = A ln|x - a| + C
∫ A/(x - a)² dx = -A/(x - a) + C
∫ 1/(x² + a²) dx = (1/a) arctan(x/a) + C
U-sub shortcut for rational functions
Before decomposing, always check: is the numerator a constant multiple of the derivative of the denominator? If so, the integral is just a logarithm via direct u-sub, no partial fractions needed.
Students jump into partial fractions without checking whether the fraction is proper. If deg(P) ≥ deg(Q), you must do long division first. Skipping this step produces wrong decompositions.
Students forget to check for the u-sub shortcut (numerator = multiple of denominator's derivative). This saves significant time on exams.
When factoring Q(x), students sometimes stop at a quadratic that could be factored further. Factor completely before decomposing.
Sign errors when solving for constants are the number-one source of lost marks. Double-check by plugging your constants back in and verifying the original fraction.
⚠️ Partial fractions problems are almost guaranteed on the midterm and final. They test both algebraic skill and integration fluency.
⚠️ Expect at least one problem where the correct first step is long division, not decomposition.
⚠️ Strategy-selection problems ("evaluate this integral") that do not tell you which technique to use are common. Practise the decision tree until it is automatic.
Q: Evaluate ∫₂⁵ (x² + 4)/(x - 1) dx.
A: Long division gives (x² + 4)/(x - 1) = x + 1 + 5/(x - 1). Integrating: [x²/2 + x + 5 ln|x - 1|]₂⁵ = 7/2 + 5 ln 2.
Q: Evaluate ∫ (12x³ - 81x² + 174x - 117)/(x⁴ - 9x³ + 29x² - 39x + 18) dx.
A: The numerator is 3 times the derivative of the denominator. So I = 3 ln|x⁴ - 9x³ + 29x² - 39x + 18| + C.
Q: You see ∫ x² e^x dx on an exam. Which technique do you use and why?
A: Integration by parts (twice). The integrand is a product of an algebraic function (x²) and an exponential. By LIATE, u = x², dv = eˣ dx.
Q: You see ∫ √(9 + 4x²) dx. Which technique?
A: Trig substitution. The form is a² + b²x² with a = 3, b = 2. Set x = (3/2) tan θ.
Q: You see ∫ (3x + 2)/(x² + x - 6) dx. Which technique?
A: Factor the denominator: x² + x - 6 = (x + 3)(x - 2). Check whether 3x + 2 is a multiple of the derivative 2x + 1. It is not, so use partial fractions: A/(x + 3) + B/(x - 2). Solve for A and B, then integrate to get logarithms.
Q: What should you always check before starting partial fraction decomposition?
A: Two things. First, is the fraction proper (deg numerator < deg denominator)? If not, do long division. Second, is the numerator a constant multiple of the derivative of the denominator? If so, the answer is a logarithm via direct u-sub, no decomposition needed.
Partial fractions connects to algebra (factoring, polynomial division) and to logarithmic and inverse-trig integration. It also appears in Laplace transforms, which are central to differential equations and control systems.
The strategy-selection skill you build here carries directly into later courses. In multivariable calculus and differential equations, you will routinely face integrals that require you to identify the right technique on sight.
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