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WASpecialist MathematicsSyllabus dot point

How do we split a rational function into simpler fractions so that each piece integrates to a logarithm?

Resolve a rational function into partial fractions and integrate each term

WACE Specialist Unit 4 partial fractions: decomposing a proper rational function over distinct linear factors, solving for the constants, integrating each term to a logarithm, and handling improper fractions by division first, with a worked example.

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  1. What this dot point is asking
  2. When and how to decompose
  3. Improper fractions: divide first
  4. Integrating the pieces

What this dot point is asking

SCSA wants you to decompose a rational integrand into partial fractions and integrate, including reducing an improper fraction by division first.

When and how to decompose

Partial fractions apply to a proper rational function (numerator degree less than denominator degree) whose denominator factors. For distinct linear factors, set up one constant over each factor:

Multiply through by the denominator and solve for the constants, either by substituting the roots x=ax = a and x=bx = b (the cover-up method) or by equating coefficients.

Improper fractions: divide first

Integrating the pieces

Each term Ax−a\tfrac{A}{x - a} integrates to Aln⁡∣x−a∣A\ln|x - a|. So the integral of a decomposed rational function is a sum of logarithms (plus any polynomial part from division), combined into a single logarithm where convenient using log laws.