Technique Masterclasses

Pure Mathematics 1

Deep Dive: Integrate Twice Two Constants

CODEX Deep-Dive: Integrate Twice, Two Constants (WMA11, spec 4.1 / 5.1 / 5.2)

Provenance (2026-09-29). Built from a direct read of research/veridian/WMA11-verified-facts.md — specifically §1 (spec table of contents), §4.5 (Integration), §6 ("topics with enough verified material"), and the full 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 Jan 2023 and Oct 2021 mark-scheme PDFs, not a trust of any earlier paraphrase); the existing lesson content/veridian/curriculum/wma11/integrate-twice-two-constants.ts, read in full; the existing signal-topic-brief-integrate-twice-two-constants.md and codex-formula-and-method-bank.md §1.5 / §2.5 in this same folder, both checked for overlap before writing a line of this document; and content/veridian/question-bank/wma11.ts, grepped and confirmed to contain no item tagged to this lesson. No new primary-source PDF was opened for this pass. Not affiliated with or endorsed by Pearson Edexcel.

0. What this document adds, and what it deliberately does not repeat

The lesson already carries a complete worked chain, chain-drill and marked-solution built on VERIDIAN-original, clean-integer f″(x) expressions, chosen specifically because the one real fully-verified instance of this exact question type does not resolve to a clean expression once only its two real conditions are kept. The SIGNAL brief already gives the frequency count (1 of 5 series fully mined for the two-constant structure, 2 more for a single-constant sibling) and a mark-tariff comparison. The formula bank already gives the compressed two-line procedure and the one governing citation.

This document adds five things none of the three above states:

  1. The real Jan 2023 Q11(b) numbers, worked through in full — the actual f″(x), the actual arithmetic, and the actual final f(x), none of which appears anywhere else in this repository outside the raw audit-log entry.
  2. A general algebraic argument, independent of any specific example, for why the first constant of integration always resurfaces as a genuine linear term of f(x) — not just true for the numbers the lesson happens to use, but true for any f″(x) whatsoever.
  3. A real, distinct exam angle: in the one fully-verified instance, neither condition was handed over as a bare number — both were extracted from a single tangent-line fact.
  4. A third kind of unknown, confirmed on an independent real question, that looks like a constant of integration but structurally is not one, and is marked by a different convention as a result.
  5. A synthesis of the mark-dependency architecture stated precisely enough to explain which errors are recoverable and which are not — not just that the marks exist, but why the dependency pattern is shaped the way it is.

Where the real record runs out, each section below says so rather than continuing.

1. Where "integrate twice, two constants" actually sits in the spec

There is no single spec bullet named "integrate twice." The question type is three spec items composed: 4.1 gives second-order derivatives standing (f′ and f″ are both legitimate objects); 5.1 requires a constant every time indefinite integration reverses a differentiation, not once per question; 5.2 names the one-integration version explicitly — given f′(x) and a point, find y = f(x) — and says nothing about a two-integration version because it does not need to: composing 5.1 twice, once for each of the two reversals f″→f′ and f′→f, is what produces it. The rule itself, ∫xⁿ dx = xⁿ⁺¹/(n+1) + c for n ≠ −1, is not printed in the exam formula booklet and must be recalled from memory, same as the differentiation rule it reverses. The n = −1 exclusion is a genuine, checkable boundary: ∫x⁻¹ dx = ln|x| + c requires logarithms, which are P2 content, so no valid P1 question in this family can ever require integrating a term whose index is exactly −1.

2. The general mechanism: two integrations, two structurally different unknowns

The lesson already establishes the definitional split — a gradient condition f′(a) = m is a statement about f′(x) and can only be used once f′(x) exists; a point (p, q) is a statement about f(x) and can only be used once f(x) exists. What is worth adding is the algebra that makes the consequence of getting this wrong precise, for any f″(x) at all, not only the specific numbers in any one worked example.

Let F(x) be any particular antiderivative of f″(x) — whatever it happens to look like. Then f′(x) = F(x) + c₁. Integrating again:

f(x) = ∫[F(x) + c₁] dx = ∫F(x) dx + c₁x + c₂

The term c₁x falls straight out of ∫c₁ dx = c₁x, regardless of what F(x) itself is — a linear, a fractional-power, or a mixed expression all behave identically here, because the constant c₁ is added on separately from F(x) before the second integration ever happens. This is the general fact behind the specific worked examples: c₁ always resurfaces as exactly a linear term of f(x), never absorbed into c₂ and never vanishing, no matter the shape of the original f″(x). A single constant added only after both integrations are complete has no linear term to contribute at all, so it cannot reconstruct this expression — not as a matter of tidiness, but because it is missing a term the correct f(x) genuinely has. c₂, by contrast, always stays a bare additive constant; it is the only one of the two unknowns that behaves the way "the constant of integration" is informally imagined to behave.

This is also the algebra behind an asymmetry in the real mark scheme worked through in §4: the method mark for the second integration checks only that some antiderivative-shaped expression has been reached, so it survives a wrong c₁ untouched — but any mark checking the final, specific f(x) is sensitive to c₁, because c₁ is a real coefficient sitting inside a real term of the answer, not a decoration on it.

3. Real distinct sub-cases confirmed in the exam record

3.1 Clean integer-power f″(x)

This is the shape every worked example in the lesson itself uses (f″(x) = 6x; f″(x) = 12x − 6; f″(x) = 6 − 6x), chosen because it lets both real conditions resolve to a checkable answer without the original question's own f″(x), which is not preserved in the facts bank. It is not, however, the shape of the one real fully-verified instance of this exact question type — see 3.2.

3.2 A mixed integer/fractional-power f″(x) — the real, verified instance

Jan 2023 Q11(b) (Publications Code WMA11_01_MS_2301, QP Log Number P72066A) sets f″(x) = 4x + x^(−1/2), with f′(4) = 3 and f(4) = 16. Integrating once: ∫4x dx = 2x², and ∫x^(−1/2) dx = x^(1/2)/(1/2) = 2x^(1/2), so f′(x) = 2x² + 2x^(1/2) + c₁. Substituting f′(4) = 3: 2(16) + 2(4^(1/2)) + c₁ = 32 + 4 + c₁ = 3, giving c₁ = −33, so f′(x) = 2x² + 2x^(1/2) − 33. Integrating a second time: ∫2x² dx = (2/3)x³, ∫2x^(1/2) dx = 2 · x^(3/2)/(3/2) = (4/3)x^(3/2), and ∫−33 dx = −33x, giving f(x) = (2/3)x³ + (4/3)x^(3/2) − 33x + c₂. Substituting f(4) = 16, using 4^(3/2) = 8: (2/3)(64) + (4/3)(8) − 132 + c₂ = 128/3 + 32/3 − 132 + c₂ = 160/3 − 132 + c₂ = 16, giving c₂ = 284/3. The finished curve is f(x) = (2/3)x³ + (4/3)x^(3/2) − 33x + 284/3. Every step here has been independently recomputed for this document and agrees exactly with the audit-log entry's stated intermediate values (c₁ = −33) and final answer (284/3).

The genuinely additive sub-case this reveals is that the second, fractional-power term needs the identical rule from §1 applied to a non-integer, negative index — a real step the polynomial-only worked examples in the lesson never require, and one that sits close to, but legitimately on the allowed side of, the n = −1 exclusion boundary named in §1.

3.3 Conditions arrived at indirectly, not handed as two raw numbers

The most distinctive real feature of this one verified instance is that neither f′(4) = 3 nor f(4) = 16 was given to the candidate as a bare number. Both came from a single shared fact: a tangent line to the curve at the point where x = 4, given elsewhere in the question as y = 3x + 4. Its gradient, 3, is directly f′(4). Its value at x = 4 — 3(4) + 4 = 16 — is f(4), because a tangent line touches the curve at its point of contact, so the curve and the tangent share a height there even though they do not share a gradient anywhere else. This is a real exam angle distinct from anything the sequencing logic in §2 covers: before sorting which condition belongs to which function, a real question may first require recognising that a single given fact (a tangent or normal equation) encodes two different pieces of information about two different functions, rather than presenting a gradient value and a point as two separately labelled quantities.

3.4 The single-integration sibling, and a third, different kind of unknown

Oct 2021 Q10 and Jan 2025 Q6(b) confirm a simpler sibling of the same skill: one integration, one point, one constant. Oct 2021 Q10 also confirms something the two-constant structure does not raise on its own: part (a) of that same question asks for an unrestricted parameter inside f″(x)/f′(x) itself, found from a condition unrelated to any constant of integration. The real mark scheme for part (b) allows a candidate who never resolved that parameter numerically to still earn the integration and point-substitution marks by carrying the unresolved symbol through consistently — a follow-through convention, distinct from the strict method-mark dependency chain in §4 that governs c₁ and c₂ within this topic itself.

The distinction is worth stating plainly because it is easy to blur: a parameter given inside f″(x) that must be recovered from a separate, earlier condition is not one of "the two constants of integration" this lesson is about, even though it is also an unknown number sitting in the same expression. Losing marks on it is a different failure from misapplying the gradient/point sequencing this document and the lesson both cover, and the two should not be treated as interchangeable risks.

4. The real mark-dependency architecture

Jan 2023 Q11(b)'s own mark scheme, for the two-constant structure, runs six steps: a method mark for the first integration (at least one index correctly raised); a dependent method mark for substituting the gradient condition and solving for c₁ (dependent on the first mark only); an accuracy mark for the resulting f′(x); a second method mark for integrating again, reaching the correct family of shape — dependent on the first integration mark only, so it survives even a wrong or missing c₁; a triply-dependent method mark for substituting the point and solving for c₂, dependent on every method mark before it; and a final accuracy mark for the finished f(x).

The asymmetry this produces is the practical payoff of §2's algebra: the second integration's own method mark is protected from an earlier mistake in c₁, because it only checks that some valid antiderivative-shaped expression has been reached — but the mark for finding c₂ is not protected, because it depends on every step that came before it, and the final accuracy mark is not protected either, because c₁ is a real coefficient inside the finished expression, not a value that happens to not matter any more. One error early in the chain therefore blocks everything from the point-substitution step onward, even though the method mark for the second integration itself survives untouched. In the one fully reconstructed series where a whole-paper Integration allocation exists (Jan 2023), this six-mark structure is the larger of that paper's two Integration questions, sitting alongside a separate five-mark single-integration question — one data point, not a general ratio.

5. Real exam-angle synthesis

The full two-integration, two-constant structure is confirmed in exactly 1 of 5 WMA11 series fully mined question-by-question for this course (Jan 2019, Oct 2020, Oct 2021, Jan 2023, Jan 2025): Jan 2023 Q11(b). Two further series are confirmed as genuine WMA11 mark schemes but were not mined topic-by-topic (Oct 2022, Jan 2024), and two more were checked for one unrelated question each (June 2022, June 2023) — their silence on this topic is a gap in what has been checked, not a confirmed absence. The narrower single-integration sibling is confirmed in 2 of the same 5 series: Oct 2021 Q10(b) and Jan 2025 Q6(b), the latter used only as a light triangulation point and not mined to mark-code depth. Both confirmed instances sit as the last, or second-to-last, question on their respective papers.

The confirmed traps, each attributable to a named series and question: substituting the point where the gradient condition was needed, or the reverse (Jan 2023 Q11b); integrating all the way through to f(x) before applying either condition, collapsing two constants into one (Jan 2023 Q11b); transposing or sign-flipping the given point's coordinates (Oct 2021 Q10); and omitting the constant of integration outright, the single most repeated integration slip found anywhere in the WMA11 corpus reviewed for this course, independently confirmed across four series — Oct 2021 Q1, Oct 2020 Q9, Jan 2023 Q3, and Jan 2025 Q6(b) — though only the last of those four belongs to this specific "point, then find f(x)" family; the other three are plain single-integration questions elsewhere on their papers, so this particular trap is corpus-wide rather than unique to this topic.

6. What is honestly still unverified

Jan 2025 Q6(b) has not been mined to mark-code depth; nothing beyond the omitted-constant trap is confirmed about it. Oct 2021 Q10 part (a)'s own f″(x)/f′(x) expression and the numeric value of its unrelated parameter are not preserved anywhere in the facts bank, so §3.4 states the mechanism without inventing the missing numbers. Oct 2022 and Jan 2024 are confirmed genuine WMA11 series but have not been checked for this topic at all. No likelihood percentage or grade boundary appears anywhere above; every frequency claim in this document is stated as "seen in N of M series checked," per this course's standing discipline, and no further quantitative claim should be read into it.

7. Sources