Math · Calculus II · Concept
Integration by partial fractions
Partial fraction decomposition rewrites a proper rational function P(x)/Q(x) as a sum of simpler fractions, one for each factor of the denominator. Each piece has a standard integral: A/(x − a) gives A ln|x − a|, a repeated factor gives a power of 1/(x − a), and an irreducible quadratic gives a logarithm and an arctangent. If the numerator’s degree is not lower than the denominator’s, divide first.
When to use partial fractions
Use them for a rational function whose denominator factors and whose integral is not a quick substitution. The fraction must be proper, with the numerator of lower degree; if it is not, polynomial division first gives a polynomial plus a proper fraction.
One term for each factor
Factor the denominator completely into linear factors and irreducible quadratics, then write one term per factor.
| Factor of the denominator | Terms in the decomposition |
|---|---|
| x − a | A/(x − a) |
| (x − a)² | A/(x − a) + B/(x − a)² |
| x² + b², irreducible | (Bx + C)/(x² + b²) |
Find the constants
Multiply both sides by the denominator to clear the fractions. Then substitute convenient values of x, such as the zero of each linear factor, or match the coefficients of like powers of x. Substituting a zero is the “cover-up” shortcut.
Integrate each term
A/(x − a) integrates to A ln|x − a|, and A/(x − a)² to −A/(x − a). Over x² + b², split the numerator: x/(x² + b²) integrates to ½ ln(x² + b²), and 1/(x² + b²) to (1/b) arctan(x/b).
Check by recombining or differentiating
Adding the partial fractions over a common denominator should return the original fraction, and differentiating the answer should return the integrand. Either check catches a wrong constant.
Common mistakes
- Decomposing an improper fraction without dividing first.
- Giving a repeated factor only one term: (x + 1)² needs A/(x + 1) + B/(x + 1)².
- Putting only a constant over an irreducible quadratic: it needs Bx + C.
- Dropping the absolute value in ln|x − a|, which matters wherever x − a < 0.
Key terms
- Partial fraction decomposition
- Rewriting a proper rational function as a sum of simpler fractions, one for each factor of its denominator: A/(x − a) for a linear factor, an extra term for each power of a repeated factor, and (Bx + C)/(x² + bx + c) for an irreducible quadratic.
- Rational expression
- A fraction whose numerator and denominator are polynomials. Any value that makes the original denominator zero stays excluded, even if that factor cancels when you simplify.
- Irreducible polynomial
- A polynomial that can’t be factored into lower-degree polynomials with the allowed coefficients: x² + 1 is irreducible over the real numbers but not over the complex numbers.
- Natural logarithm
- The logarithm with base e, written ln x. It undoes eˣ: ln(eˣ) = x for every x, and e^(ln x) = x for x > 0.
- Inverse tangent
- arctan x, or tan⁻¹ x: the angle between −π/2 and π/2 (−90° to 90°) whose tangent is x. It is defined for every real x.
- Antiderivative
- A function whose derivative is the given function, such as x³ for 3x². Any two antiderivatives on one interval differ by a constant, which is why answers carry + C.
Work through an example
Find ∫(5x − 1)/(x² − x − 2) dx.
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