Integration by partial fractions is a method for integrating rational functions, which are fractions made from polynomials. It matters because many rational integrals are too complicated to integrate directly, but become simple after being split into smaller fractions. The main idea is to rewrite one difficult fraction as a sum of simpler fractions whose antiderivatives are familiar.
This often turns a hard problem into a sequence of logarithms and basic power rules.
The method starts by factoring the denominator and choosing a partial fraction form based on those factors. Distinct linear factors get terms like A/(x - a), repeated linear factors get A/(x - a) + B/(x - a)^2, and irreducible quadratic factors get linear numerators like (Ax + B)/(x^2 + px + q). After solving for the unknown constants, integrate each simpler term separately.
Before decomposing, always check that the rational function is proper, meaning the degree of the numerator is less than the degree of the denominator.
Key Facts
- Partial fractions apply to rational functions P(x)/Q(x), where P and Q are polynomials.
- A rational function is proper if degree(P) < degree(Q); if not, use polynomial division first.
- For distinct linear factors, P(x)/[(x - a)(x - b)] = A/(x - a) + B/(x - b).
- For repeated linear factors, include every power: A/(x - a) + B/(x - a)^2 + ... + K/(x - a)^n.
- For an irreducible quadratic factor ax^2 + bx + c, use a numerator of the form Ax + B.
- Basic integral: ∫ A/(x - a) dx = A ln|x - a| + C.
Vocabulary
- Rational function
- A function that can be written as a quotient of two polynomials, such as P(x)/Q(x).
- Proper rational function
- A rational function whose numerator has lower degree than its denominator.
- Partial fraction decomposition
- The process of rewriting one rational expression as a sum of simpler rational expressions.
- Distinct linear factor
- A first degree factor such as x - a that appears only once in the denominator.
- Repeated factor
- A factor that appears with a power greater than 1, such as (x - 2)^3.
Common Mistakes to Avoid
- Skipping polynomial division for an improper fraction, which is wrong because partial fractions require the numerator degree to be less than the denominator degree.
- Leaving out powers of a repeated factor, which is wrong because (x - a)^3 needs separate terms for (x - a), (x - a)^2, and (x - a)^3.
- Using a constant numerator over an irreducible quadratic factor, which is wrong because factors like x^2 + 1 require a numerator Ax + B.
- Forgetting absolute value in logarithmic answers, which is wrong because ∫ 1/(x - a) dx = ln|x - a| + C on intervals where the expression is defined.
Practice Questions
- 1 Compute ∫ 5/[(x - 1)(x + 4)] dx by partial fractions.
- 2 Compute ∫ (3x + 7)/(x^2 + 5x + 6) dx using partial fractions.
- 3 Explain why ∫ (x^2 + 1)/(x - 1) dx should not start with partial fraction decomposition before another algebraic step.