Pure Mathematics 3

Topic Brief: Integration By Recognition

Provenance: written 2026-09-29. Source: research/veridian/WMA13-verified-facts.md (§1 spec table; §5 "5. Integration"; §9 lesson-audit log entry for wma13-integration-by-recognition), grepped specifically for "recognition" and spec 5.1/5.2, plus this folder's own signal-frequency-and-boundaries.md (§1 frequency table), codex-deep-dive-integration-by-recognition.md (§3 real past-paper evidence) and codex-formula-and-method-bank.md (§5.2 mark-ladder note) — read in full first so this brief adds question-number and mark-tariff synthesis those documents don't already state in one place, rather than repeating their citations or their quoted examiner-report prose. content/veridian/curriculum/wma13/integration-by-recognition.ts and content/veridian/question-bank/wma13.ts were also checked; no full worked exemplar from either is reproduced here. No new primary-source PDFs were opened for this document — every citation below traces back to one of the files just named.

#SIGNAL Topic Master Brief: Integration by Recognition (WMA13, spec 5.2)

#Scope

This covers the same ground as the lesson integration-by-recognition.ts: recognising $\int f'(x)[f(x)]^n\,dx = \frac{[f(x)]^{n+1}}{n+1}+c$ and $\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+c$ , plus the spec guidance's own named examples (tan x, sec²2x) and the identity-rewrite skill for squared trig functions (sin²x, tan²x, cos²3x) that the guidance groups alongside recognition even though it isn't the same mechanism. It excludes spec 5.1 (plain standard integration with no recognition step) except where a real question mixes the two.

#Every real series that has tested this topic

13 WMA13 series have been checked in the underlying research archive. Of those 13, spec 5.2 — a question or sub-part genuinely requiring the recognition step, not just standard 5.1 integration — has a located, citable anchor in 4: October 2020, October 2021, January 2022, June 2023. That is five real questions/sub-parts across those four series.

Series	Q	Pattern	What it actually tested
Oct 2020	Q9(c)	Log-form	Modulus-drop trap only — full question text not transcribed in the archive
Oct 2021	Q5(i)	Power-form, negative $n$ , definite	$\int_2^4\frac{8}{(2x-3)^3}\,dx$ — 4 marks
Oct 2021	Q5(ii)	Power-form, indefinite	$\int x(x^2+3)^7\,dx$ "in simplest form" — 2 marks
Jan 2022	Q3(i)	Power-form, indefinite	$\int(2x-5)^7\,dx$ — 2 marks
Jan 2022	Q3(ii)	Log-form, coefficient mismatch, definite	$\int_0^{\pi/3}\frac{4\sin x}{1+2\cos x}\,dx=\ln a$ — 4 marks
Jun 2023	Q9(c)	Booklet-derivative-in-reverse	$\int\csc x\cot x\,dx$ , credited via the printed $\csc x\to-\csc x\cot x$ read backwards — not $\int\csc x\,dx$ itself (`codex-formula-and-method-bank.md`'s own correction of the source material's earlier mischaracterisation)

#Where in the paper it lands

Question numbers above run Q3, Q5 and Q9(c) — there is no fixed slot, and the sample is too small (four series) to claim a tendency beyond that. Two real shapes recur, though: when the recognition integral stands alone, it sits at Q3 or Q5, worth 2–4 marks on its own. When it appears as Q9(c), it is nested inside a longer, trig-identity-heavy question rather than being the whole of Q9 — its own isolated mark share within that question is not separately stated in the archive for either Oct 2020 or Jun 2023, and is not invented here.

#Mark-tariff pattern

A recognition antiderivative on its own, indefinite: 2 marks, M1 A1. Confirmed twice — Oct 2021 Q5(ii), Jan 2022 Q3(i). M1 is generous (any constant, correct or not, in the right form — power raised by one, or log of the bracket); A1 requires the exact constant AND the $+c$ together, as one combined mark. There is no separate dM1 for "get the constant right" within one antiderivative — the source material's own audit checked this explicitly across all three same-shaped mark schemes it holds and found the two-mark M1/A1 split every time.
The same antiderivative inside a definite integral: 4 marks, M1 A1 dM1 A1. Confirmed twice — Oct 2021 Q5(i), Jan 2022 Q3(ii). The extra dM1 A1 pair is reserved for substituting both limits and subtracting — a genuinely separate, later step, not a re-split of the antiderivative mark. On an indefinite integral there is "no requirement for the +c" language once the definite-integral limits step is what's being marked; on the antiderivative step itself the $+c$ is required.
The Q9(c) anchors (Oct 2020, Jun 2023) have no isolated mark tariff stated in the archive for the recognition sub-step, since both sit inside a longer multi-part question. Treat the 2-mark / 4-mark pattern above as applying where the recognition integral is its own question part.

#Recurring context/angle

Two real shapes, not one:

Power-form recognition, $\int f'(x)[f(x)]^n\,dx$ with a linear or simple polynomial $f(x)$ — confirmed three times: Oct 2021 Q5(i) (negative $n$ , $f$ linear), Oct 2021 Q5(ii) ( $f(x)=x^2+3$ ), Jan 2022 Q3(i) ( $f$ linear). All three are the "generous M1 on form, exact A1 on constant+c" pattern above.
Log-form recognition, $\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+c$ , sometimes with a coefficient mismatch between the numerator and the true $f'(x)$ requiring an adjusting constant — confirmed on Jan 2022 Q3(ii) ( $f(x)=1+2\cos x$ , numerator $4\sin x$ is $-2\times f'(x)$ ) and, in modulus-trap form only, Oct 2020 Q9(c). Jun 2023 Q9(c) is a related but distinct case — recognising a booklet-printed derivative run backwards, not deriving a general- $f$ log form from scratch.

Neither shape has been located tested as the entire question — every real anchor above is either a standalone 2–4-mark sub-part or one sub-part of a larger question.

#If this topic appears, here's what to expect

Expect a short (2- or 4-mark) sub-part, most likely at Q3 or Q5, or nested inside a longer, identity-heavy Q9 — not the whole question.
If it's an indefinite integral asked to be given "in simplest form," expect 2 marks total: one for reaching the right form with any constant, one combined mark for the exact constant together with $+c$ . Do not expect a separate mark just for spotting the pattern versus getting the constant right.
If it's a definite integral, expect 4 marks total — the antiderivative (M1 A1) plus a separate, dependent pair (dM1 A1) for substituting and subtracting the limits.
Keep the modulus on any log-form antiderivative until the specific $f(x)$ on that question is known to be positive over the given domain — the two log-form anchors located (Oct 2020, and independently the identical error confirmed on Jun 2023 elsewhere in this paper's own log-integration content) both turn on this exact drop.
If the integrand is $\csc x$ -related, check whether the needed derivative is already printed in the formula booklet's own P3 section and can be read in reverse before assuming a derivation or substitution is required from scratch.
Do not default to a log-form antiderivative "to be safe" on a power-form integrand with no $[f(x)]^{-1}$ factor in it, or vice versa — a real, confirmed confusion in this topic's examiner-report material (cited in full in codex-deep-dive-integration-by-recognition.md §3, not repeated here).

#What wasn't found

No series in the archive tests spec 5.2 as the entirety of a question — every real anchor is a sub-part. Oct 2020 Q9(c)'s full question text is not transcribed anywhere in the archive, only its modulus-drop trap; that gap is stated here rather than filled with an invented question. Five of the paper's 13 series were checked for other specific topics or citation-currency only, so "4 of 13" is what has been verified so far, not a claim that spec 5.2 is provably absent from every other series. No likelihood or percentage claim is made anywhere above; every count is "seen in N of the M series checked."

#Sources

research/veridian/WMA13-verified-facts.md; signal-frequency-and-boundaries.md, codex-deep-dive-integration-by-recognition.md, codex-formula-and-method-bank.md (this folder).