Topic Brief: Partial Fractions Decomposition and Integration

Provenance: compiled 2026-09-29. Sources checked directly for this document: research/veridian/WMA14-verified-facts.md §5.2 (this topic's own three real citations — the entire evidentiary base), §6 (series-reviewed methodology and the "N of M series" denominator convention), §7 (the topics-with-enough-material-for-a-lesson list) and §8 (flagged gaps); the header and prerequisites/sourcing-discipline note of content/veridian/curriculum/wma14/partial-fractions-decomposition-and-integration.ts (checked to confirm which quotes are already used there, so nothing below repeats a worked exemplar); content/veridian/technique-hub/wma14/codex-deep-dive-partial-fractions-decomposition-and-integration.md's own header and scope note (a theory/proof document, not a frequency one — checked to confirm no overlap); signal-frequency-and-boundaries.md §1's existing row for this topic (the "3 of 7" figure and its "no single confirmed total" note, both reused directly rather than re-derived); and forge-exam-technique-guide.md §1's mark-tariff table and its algebraic-manipulation checklist entries for this topic. content/veridian/question-bank/wma14.ts was checked directly and contains no entry for this topic. No fresh PDF extraction was performed this session — every fact below reuses citations already verified elsewhere in this facts bank. Not affiliated with or endorsed by Pearson Edexcel.

#SIGNAL Topic Master Brief — Partial Fractions Decomposition and Integration (spec 2.1, 6.3)

#Frequency: seen in 3 of 7 series checked

Of the 11 WMA14 series with at least one primary document in the facts bank, only 7 received real, question-by-question mining rather than a boilerplate-only mark-scheme check: January 2021, October 2021, June 2022, October 2022, January 2024, October 2024, January 2026 (signal-frequency-and-boundaries.md §1). Three of those seven carry a verified citation for this topic — October 2021 (Q3(a)), October 2022 (Q2) and January 2024 (Q2) — all three sourced from full Examiners' Reports, which is itself worth noting: this topic's evidence base is examiner-report-only, with no mark-scheme-only sighting adding a fourth series. January 2021, June 2022, October 2024 and January 2026 show no citation for this topic in this course's review; per the standing convention in this facts bank, that means no finding was captured for those series, not that the topic was confirmed absent from those papers.

#Mark-tariff pattern: no confirmed total, one partial data point

Unlike several other WMA14 topics, this one has no single confirmed mark-tariff total anywhere in this course's source review (signal-frequency-and-boundaries.md §1; absent from the real-question table in forge-exam-technique-guide.md §1). The closest thing to a number on record is qualitative: the October 2022 report calls its own question "extremely well done with the majority of candidates scoring at least 5 marks" — a floor, not a total, and not broken down by part. Structurally, two of the three citations are explicitly part-lettered — January 2024's Q2(a) (decomposition) and Q2(b) (the resulting integral) — which is the only structural signal available: when this topic is examined, it tends to be split into a decomposition part and a separate part that applies the result, most often to integration, rather than set as one undivided prompt.

#Context: decomposition paired with its own integration, and a method-choice angle

The clearest recurring pattern across the three series is not a context or scenario type (there is no real-world wrapper in any of the three citations — every one is a bare algebraic/calculus prompt) but a pairing of spec points within one question. October 2022 Q2 tests decomposition feeding directly into integration in a single question (spec 2.1 into 6.3), and January 2024 Q2(a)/(b) does the same across two lettered parts. Two of the three series with a citation for this topic (October 2022, January 2024) show this pairing; October 2021's citation (Q3(a)) covers only the decomposition part, with no companion integration part captured in this review.

A second real, cited pattern is method choice being an explicit examiner-report topic, not just a students'-preference footnote. Two legitimate decomposition routes are named across the three series: substitution (setting $x$ to convenient values) versus full simultaneous-equation expansion of matched coefficients (January 2024 Q2(a)), and the identity method versus long division (October 2021 Q3(a)). In both cases the report records one route as more error-prone than the other — simultaneous equations "more prone to error than attempts via substitution," and long division specifically flagged for a sign slip in its remainder — which makes method choice itself part of what this topic tests, not incidental to it.

The single most robust finding here is a two-series-confirmed recurring error, not a one-off: omitting the $\frac1a$ scaling factor when integrating $\int\frac1{ax+b}\,dx=\frac1a\ln|ax+b|+c$. October 2022's report records it as "a common error… to omit the ½ in $\int\frac1{2x-1}\,dx=\frac12\ln(2x-1)$," and January 2024's report records the same slip independently on its own question ("a number of students failed to divide the second logarithm by two"). Students correctly recall the $\ln$ form itself; the failure is specifically the reciprocal-of-coefficient scaling term.

#What is, and isn't, on record

Two distinct, real errors are confirmed on the decomposition side, but each from a single series only, so neither should be read as a general pattern the way the $\frac1a$ scaling slip above is: October 2021's over-multiplication by an extra power of the repeated linear factor ("$(2x+1)^3$" instead of the correct product of denominators), and the same series' long-division sign error (setting a numerator to $+6$ instead of $-6$ and losing the sign when substituting back). Neither error recurs in the October 2022 or January 2024 citations. No finding in this facts bank ties this topic to spec 4.1's binomial-series application (the spec's own guidance names partial fractions as a route into series expansion of rational functions), to differentiation, or to any real-world context — the three citations are pure algebra-and-calculus prompts. No grade-boundary data exists anywhere in this course's WMA14 source review, so none is implied here.

#If this topic appears: what to expect

On the three confirmed data points available: expect a question built from a rational function with a linear or repeated-linear denominator (the spec explicitly excludes irreducible quadratic factors), most likely split into a lettered decomposition part followed by a part that uses the result — commonly integration, per the spec's own 6.3 cross-reference and two of the three real citations. Decide the decomposition method (substitution vs. simultaneous equations for the constants; identity vs. long division for a top-heavy numerator) before starting rather than defaulting to one out of habit, since Pearson's own reports treat that choice as a real source of marks gained or lost. On any resulting $\int\frac1{ax+b}\,dx$ term, write the $\frac1a$ factor down explicitly before integrating — the single most-repeated, two-series-confirmed slip on this topic is losing exactly that term. Treat "3 of 7" as a genuinely thinner evidence base than this course's better-covered topics (proof by contradiction sits at 6 of 7), not as licence to under-revise a topic the spec lists as compulsory.