Topic Brief: Integrate Twice Two Constants

Provenance: compiled 2026-09-29. Source is research/veridian/WMA11-verified-facts.md — specifically §4.5, §6, and the integrate-twice-two-constants.ts entry in the "Lesson-audit log" (§8, dated 2026-09-14, itself a fresh pdftotext -layout/-raw re-extraction of the primary mark-scheme PDFs, not a trust of the facts bank's earlier paraphrase) — cross-checked against content/veridian/curriculum/wma11/integrate-twice-two-constants.ts and this folder's codex-formula-and-method-bank.md, forge-exam-technique-guide.md and signal-frequency-and-boundaries.md to confirm this brief adds a genuinely new layer rather than restating theirs. No fresh PDF extraction was done for this document itself. Not affiliated with or endorsed by Pearson Edexcel.

#SIGNAL Topic Master Brief — Integrate Twice, Find Two Constants (spec 4.1, 5.1, 5.2 combined)

#Frequency: seen in 1 of 5 series checked — plus a closely related variant in 2 more

The facts bank fully mined five WMA11 examiner reports question-by-question: Jan 2019, Oct 2020, Oct 2021, Jan 2023, Jan 2025. The exact structure this topic teaches — f″(x) given, integrated twice in sequence, with two independent conditions of different kinds (a gradient value for the first constant, a point on the curve for the second) — is confirmed in exactly 1 of those 5: Jan 2023 Q11(b).

Stating only "1 of 5" would understate the record, though. A narrower, single-integration sibling of the same skill — f′(x) given, integrated once, with one point condition to find one constant — is confirmed in 2 more of the 5: Oct 2021 Q10(b) and Jan 2025 Q6(b). The 2026-09-14 lesson-audit pass, which re-extracted all three mark schemes fresh from primary PDFs rather than trusting an earlier summary, is explicit that these two are "single-integration case[s]," distinct from Jan 2023 Q11(b)'s genuine two-integration, two-constant structure. An earlier line in the same facts bank (§6) groups all three together as "three independently-verified series" — true for the shared sequencing logic (a slope fact belongs to f′, a height fact belongs to f), not true that all three carry two constants. This brief keeps the counts separate: a reviser deciding how much of the double-integration structure specifically to drill needs the narrower number.

The other four series this research pass touched (Oct 2022, Jan 2024 — confirmed genuine mark schemes, never mined topic-by-topic; June 2022, June 2023 — read for one unrelated targeted question each) were not checked for this topic at all. Their silence here is a gap in the research pass, not a confirmed absence.

#The one confirmed two-constant appearance

Jan 2023 Q11(b) (Publications Code WMA11_01_MS_2301, QP Log Number P72066A) — 6 marks, the back half of an 8-mark question (part (a), a separate normal-line-geometry question, carries the other 2). f″(x) = 4x + x^(−1/2); gradient condition f′(4) = 3; the curve passes through the point found via the tangent line at that same x, giving f(4) = 16. Real mark-scheme sequence, verbatim structure: M1 (integrate f″(x) once, one index correct) → dM1 (substitute f′(4)=3, solve for the first constant — dependent on that M1) → A1 (f′(x) = 2x² + 2x^(1/2) − 33) → dM1 ("dependent upon first M1 only" — integrate a second time to reach a cubic-family shape, regardless of whether the first constant was right) → dddM1 ("dependent upon all previous M's" — substitute f(4)=16, solve for the second constant) → A1 (final f(x), exact, isw).

Two structural facts worth naming because they are specific to this question, not general integration lore: the second integration's method mark depends only on the first M1, not on the first constant being correct — a student who mishandles c₁ can still bank that mark; but the final constant-finding mark (dddM1) depends on every prior method mark, so either sequencing error (wrong-stage substitution, or deferring both constants) blocks it. The examiner report names both real failure modes directly: candidates "substituted the coordinates for P" instead of recognising that f′(4) = 3 — a fact about the gradient function, not the curve itself — was what the stage actually needed; and, separately, it reports this as "very common": candidates who "integrate twice and apply a constant only after their second integration," losing marks that were available earlier.

#The adjacent single-constant appearances

Oct 2021 Q10(b) — 4 marks: M1 (integrate) → A1ft (correct integrated form, follow-through on an earlier unknown) → dM1 (substitute the given point, solve for c) → A1 (final answer). Part (a) of the same question separately finds an unknown constant inside f″(x) itself — a different unknown, resolved before part (b) starts, which is exactly why part (b) only needs one condition and one constant rather than two. Confirmed trap: candidates transposing the given point, "using the point (1, 8) or even (−8, 1)" in place of the real (1, −8).

Jan 2025 Q6(b) — used by the facts bank only "as a third triangulation point," not mined to the same mark-by-mark depth as the other two; its main confirmed fact is that it independently repeats the corpus-wide "forgot the constant of integration" slip.

#Mark-tariff pattern

The two structures sit at different tariffs, and the gap tracks the extra dependency, not just extra arithmetic. The confirmed two-constant version (Jan 2023 Q11(b)) is 6 marks with a three-deep method-mark chain (M1 → dM1 → dM1 → dddM1); the confirmed single-constant version (Oct 2021 Q10(b)) is 4 marks with a two-deep chain (M1 → dM1). Both sit late in their papers — Q11 of 11 and Q10 of 10 respectively — worth noting with only two data points, not generalising from. On the one series with a full whole-paper mark breakdown (Jan 2023), the facts bank ties indefinite integration to two questions: Q3 (5 marks, the unrelated fraction-split question) and Q11(b) (6 marks, this topic) — so this topic accounts for just over half that series' Integration marks, 6 of 11. Treat this as one data point, not a settled ratio: it is a single fully-reconstructed series, and no other series in the facts bank has a comparable breakdown.

#The recurring angle

In the one confirmed instance, the split is always the same two kinds of fact, never two of the same kind: a gradient value at a named x (a fact about f′, usable the moment f′ exists) and a point on the curve (a fact about f, usable only once f exists). Nothing in the confirmed record shows two gradient conditions, two points, or a condition on f″(x) itself beyond the expression to be integrated. The wrong-stage error the examiner report names — candidates "substituted the coordinates for P" where the gradient condition was needed — is a direct consequence of that split: once f′(x) has been found, the point is the only other number the question gave, which is plausibly why some candidates reach for it a stage early. That last clause is this brief's own inference, not a claim the examiner report itself makes.

#If this topic appears: what to expect

Expect it late in the paper, worth roughly 6 marks on its own, built from exactly two conditions — one gradient value tied to f′(x), one point tied to f(x) — never two of the same kind. Expect the mark scheme to reward finding and fixing each constant immediately after its own integration, with the final constant-finding mark blocked by any earlier method slip but the second integration itself protected from an earlier wrong constant. The two confirmed, examiner-quoted failure modes are: substituting the point where the gradient condition was needed (or vice versa), and integrating twice before applying either condition. A single-integration version of the same logic (one condition, one constant) is confirmed more often in the wider record and carries a shorter, ~4-mark tariff with only one dependency step — worth recognising as the same skill at lower stakes, not the same question.