Topic Brief: Integration By Parts and Substitution

Provenance: compiled 2026-09-29. Sources checked directly for this document: research/veridian/WMA14-verified-facts.md §1 (spec 6.2, verbatim), §4 (general marking-guidance conventions), §5.6 (every "Integration by parts" / "Integration by substitution" citation for this spec point), §6 (series-review methodology and the 7-series question-level-mining list), §7 (the "four series" topic-summary line), §8 (flagged gaps), and the lesson-audit-log entry for integration-by-parts-and-substitution.ts dated 2026-09-14 (the June 2022 Q8(b) mark-scheme anchor); the full header of content/veridian/curriculum/wma14/integration-by-parts-and-substitution.ts (its own "SERIES-COUNT DISCIPLINE" note, cross-checked against §5.6 and the audit log directly); content/veridian/technique-hub/wma14/codex-deep-dive-integration-by-parts-and-substitution.md Parts 3–4 (read in full to avoid restating its worked micro-examples and sub-case taxonomy); the existing row for this topic in signal-frequency-and-boundaries.md §1 and its own flagged-gaps method in §3; and forge-exam-technique-guide.md §1's pacing-table row plus its line-228 "real questions cited by series" list. content/veridian/question-bank/wma14.ts was checked directly and contains no entry for this topic. No fresh PDF extraction was performed — every fact below reuses a citation already verified elsewhere in this facts bank. This is the second SIGNAL topic-master-brief written for WMA14, after area-under-a-parametric-curve; its purpose is the same "if this appears, what to expect" synthesis in one place, kept separate from CODEX's technique taxonomy and FORGE's exam-craft layer. Not affiliated with or endorsed by Pearson Edexcel.

#SIGNAL Topic Master Brief — Integration By Parts And Substitution (spec 6.2)

#Frequency: seen in 4 of 7 series checked

Of the 11 WMA14 series with at least one primary document in this facts bank, 7 received real, question-by-question mining: January 2021, October 2021, June 2022, October 2022, January 2024, October 2024, January 2026 (signal-frequency-and-boundaries.md §1). Four of those seven carry a real citation for spec 6.2: October 2021 (Q6, substitution; Q8(b), by parts), June 2022 (Q8(b), by parts — mark scheme only, no examiner report published for this series), October 2022 (Q7(i), substitution; Q7(ii), by parts — the same question testing both), and January 2024 (Q5(a), by parts). Not seen in January 2021, October 2024 or January 2026.

This corrects, rather than restates, the existing tally: signal-frequency-and-boundaries.md §1 lists this topic as "3 of 7" (Oct 2021, Oct 2022, Jan 2024 only), omitting June 2022. That anchor is real — the facts bank's own lesson-audit log used it to fix a dM1-vs-M1 marking error in the live lesson file (Question Paper Log Number P71381A, mark scheme wma14-01-rms-20220818.pdf) — and the CODEX deep-dive for this topic already counts it separately ("three examiner-report series... plus one mark-scheme-only anchor," Part 3). The lesson file's own header, read strictly against §5.6's citation list, correctly says "exactly three, never four" — but that list never absorbed the June 2022 anchor, added later in the same facts bank's lesson-audit log (2026-09-14). Treating a mark-scheme-only citation as a countable series is the convention this facts bank already applies elsewhere — the separable-differential-equations row counts a mark-scheme-only January 2026 citation the same way — so 4 of 7, not 3, is the count consistent with that precedent.

#Mark-tariff pattern: one exact figure, the rest unweighed

Only one sub-part's exact mark allocation is confirmed: June 2022 Q8(b) is worth 3 marks (M1, dM1, A1 on the traditional method; M1, dM1, A1 on the credited DI/tabular alternative), inside a 10-mark Q8. No other citation's exact tariff is recorded — the Oct 2021, Oct 2022 and Jan 2024 citations all come from examiner-report prose describing where marks were lost, not the mark scheme's own per-line allocation, so there is no second data point to check the one confirmed figure against. What is confirmed structurally, without exception, across all four series: this topic has never appeared as a whole question by itself. Every real citation is a lettered or numbered sub-part of a larger question — Q6 and Q8(b) [Oct 2021], Q8(b) [Jun 2022], Q7(i)/(ii) [Oct 2022], Q5(a) [Jan 2024] — meaning the other part(s) of each question test something else entirely.

#Context: almost always a repeated application, and often paired with substitution in the same series

Three of the four real by-parts citations state directly that the tested step is a second application: Oct 2021 Q8(b) ("a second application of parts, reusing part (a)'s result"), June 2022 Q8(b) (its dM1 is explicitly "dependent on the previous method mark"), and Oct 2022 Q7(ii) ("during a second parts application"). The fourth, Jan 2024 Q5(a), is credited via the DI/tabular method — a technique built specifically for tracking repeated, not single, applications — consistent with, though not an explicit statement of, the same pattern. On the record, no citation anywhere tests a single, one-shot application of parts in isolation.

Separately: in every series this topic is cited, substitution and by-parts appear together, but not in the same shape twice. October 2021 tests them as two distinct questions (Q6 substitution, Q8(b) parts). October 2022 tests them as two sub-parts of the same question (Q7(i) substitution, Q7(ii) parts) — and that single question shows an identical failure shape on both sides: a coefficient dropped partway through the working, "losing the factor 4 in the expression" on the substitution side and candidates who "required a factor of 3 within the final integral" on the parts side. June 2022 and January 2024 each carry only a by-parts citation, with no substitution citation confirmed for either series.

#What is, and isn't, on record

No examiner's report has been published for June 2022's series, so that citation is mark-scheme mechanics only — nothing is known about how real candidates attempted Q8(b), only what the scheme rewards. The reverse gap holds for Oct 2021, Oct 2022 and Jan 2024: rich examiner-report prose, no mark-scheme line-by-line tariff for the sub-part quoted. No grade-boundary data exists anywhere in this course's WMA14 source review (signal-frequency-and-boundaries.md §2), so none is implied here either. Two further named failure modes sit on record: dropping the substitution's derivative factor entirely — "ignored the dx and simply wrote it as du" (Oct 2021, Q6) — and using a true identity as an unjustified mid-derivation shortcut that breaks a traceable mark-scheme chain, treating ln(x²) as interchangeable with 2 ln x without a stated step (Oct 2021, Q8(b)).

#If this topic appears: what to expect

Expect it as one sub-part of a larger question, not the whole question — every real citation confirms this. If the sub-part is by parts, expect a repeated application to be the tested case: write the first application out in full before starting the second, since the second application's method mark is dependent (dM1) on the first being earned, confirmed directly from the one real mark scheme available (June 2022 Q8(b)). The DI/tabular shortcut is a genuinely Pearson-credited alternative for the repeated case, confirmed independently twice (Jan 2024 examiner report; June 2022 mark scheme). If the sub-part is substitution, do not drop the derivative factor when changing variable (Oct 2021, Q6), and if the integral is definite, convert the limits into the new variable rather than reusing the original ones (Oct 2022, Q7(i)). Where both techniques appear in the same series, expect them either as two separate questions or as two sub-parts of one combined question — both patterns are on record. Treat "seen in 4 of 7" as a gap in this course's evidence, not permission to under-revise: 6.2 is a compulsory, explicitly spec-listed pair of techniques, tested or not in any specific series this review has caught.