Deep Dive: Integration By Parts and Substitution

Provenance: written 2026-09-29. Read/checked before writing: research/veridian/WMA14-verified-facts.md in full, grepped specifically for "parts"/"substitution"/"6.2" (§1 spec table, §4 general marking guidance, §5.6, and the integration-by-parts-and-substitution.ts lesson-audit-log entry dated 2026-09-14); the live lesson file content/veridian/curriculum/wma14/integration-by-parts-and-substitution.ts in full (900 lines); both existing technique-hub documents for this paper, codex-formula-and-method-bank.md (Part 1 §6 and Part 2 §2.7) and forge-exam-technique-guide.md (cross-references only); and content/veridian/question-bank/wma14.ts (236 lines, confirmed to contain no spec-6.2 item — its only worked item is binomial expansion). No new primary PDF was fetched this pass; every exam-board fact below traces to a citation already verified in the facts bank. Every micro-example not already present in the cited lesson is new content for this document, checked by direct differentiation before being written in, in the same discipline the cited lesson already uses.

#WMA14 · CODEX Deep-Dive — Integration by Parts and by Substitution (spec 6.2)

#Why this document exists, and what it doesn't repeat

The curriculum lesson (integration-by-parts-and-substitution.ts) already derives both formulas from first principles and works four spec-named integrands in full: $\int\ln x\,dx$, $\int x\sqrt{x-2}\,dx$ (indefinite and as a definite integral with limits converted), $\int x^2e^x\,dx$ (two methods), and $\int e^x\sin x\,dx$ (the cyclic case). The formula bank (codex-formula-and-method-bank.md) already gives the compressed method skeleton. Neither is reproduced here. This document does three things neither of those does: it organises the topic as a small, closed set of real sub-cases the spec guidance and the exam evidence actually distinguish, rather than as a sequence of worked examples; it supplies short, new, independently-checked micro-examples for the sub-cases the existing lesson doesn't already illustrate in that exact shape; and it synthesises every real exam angle the facts bank has documented for this spec point into one place, organised by failure/credit category rather than by series.

#Part 1 — The real model, compressed

Spec 6.2, verbatim guidance (facts bank §1): "Simple cases of integration by substitution and integration by parts. Understand these methods as the reverse processes of the chain and product rules respectively." Both are genuinely "reverse" processes, not new rules: integrating the product rule $\frac{d}{dx}[uv]=u\frac{dv}{dx}+v\frac{du}{dx}$ across both sides and rearranging gives

$$\int u\frac{dv}{dx}\,dx = uv-\int v\frac{du}{dx}\,dx$$

— printed in the Mathematical Formulae and Statistical Tables booklet's P4 section (confirmed in the facts bank's own formula masterlist), so nothing about the formula itself is examined, only the choice of $u$ and $\frac{dv}{dx}$. Substitution has no separate booklet line — it falls out of applying the chain rule to a composite antiderivative $F(g(u))$, which is why the spec frames it as the chain rule's reverse rather than as its own formula. Both derivations are given step-by-step in the cited lesson's two mechanism blocks and are not repeated here.

The marking grammar both techniques sit inside is general to the whole paper, not specific to 6.2, but has two direct 6.2-specific consequences worth stating once, cleanly, rather than leaving implicit: "Method mark for quoting a correct formula and attempting to use it" (facts bank §4) means writing $\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx$ down before touching the specific integral is a real, scheme-endorsed safety net, not exam folklore; and a genuinely repeated application of parts earns a dependent second method mark (dM1), never an independent one — confirmed directly from a real mark scheme (June 2022, WMA14/01, Q8(b): "Dependent on the previous method mark", verbatim) and, separately, from the fact that the cited lesson's own audit trail records exactly this dependency being dropped and then corrected in three places before publication.

#Part 2 — The real distinct sub-cases

The spec's own guidance column for 6.2 names four integrands and one general statement. Read closely, they pick out six structurally distinct sub-cases, not four worked examples — the difference matters for exam prep because each sub-case has its own failure mode, and the failure modes don't transfer between sub-cases (a student fluent in the cyclic case can still lose the marks that belong to limit conversion).

#2.1 Parts — the disguised single-factor product

$\ln x$ is not a visible product, so the technique doesn't obviously apply until $\ln x$ is rewritten as $\ln x\times1$. This is the spec's own required case ("the integral $\int\ln x\,dx$ is required", facts bank §1) and is fully worked in the cited lesson. The one thing worth stating explicitly, because it generalises beyond this single example: $\ln x$ can never be chosen as $\frac{dv}{dx}$, in any product it appears in, because no elementary antiderivative of $\ln x$ exists independently of this trick — $\ln x$ is always $u$.

#2.2 Parts — the baseline visible product

Before "more than one application" makes sense as a distinct sub-case, there has to be a baseline: a single visible product of two structurally different factors, one application, no repetition. The cited lesson's MCQ bank tests this shape ($\int x\cos3x\,dx$) but doesn't work it as a reference example, so here is a short one with a different function, checked independently:

$$\int x\sin2x\,dx,\quad u=x,\ \frac{dv}{dx}=\sin2x \;\Rightarrow\; \frac{du}{dx}=1,\ v=-\tfrac12\cos2x$$ $$=-\tfrac{x}{2}\cos2x-\int\left(-\tfrac12\cos2x\right)dx=-\tfrac{x}{2}\cos2x+\tfrac14\sin2x+c$$

Checked by differentiating: $\frac{d}{dx}\left[-\frac x2\cos2x+\frac14\sin2x\right]=-\frac12\cos2x+x\sin2x+\frac12\cos2x=x\sin2x$. ✓ (VERIDIAN-original; not from any past paper.)

#2.3 Parts — repeated, terminating

Spec-named: $\int x^2e^x\,dx$, worked in full in the cited lesson (traditional method and the DI/tabular alternative, both matched against real mark-scheme wording). The general pattern behind why the table terminates, stated once rather than left implicit: for any polynomial $P(x)$ of degree $n$,

$$\int P(x)e^{kx}\,dx = e^{kx}\sum_{i=0}^{n}(-1)^i\frac{P^{(i)}(x)}{k^{i+1}}+c$$

which terminates automatically after exactly $n+1$ terms, since $P^{(n+1)}(x)\equiv0$ — this is why the DI method's D-column always reaches $0$ for a polynomial factor and never does for a trig factor (§2.4 below). This is course-original synthesis of the general pattern, not a transcribed exam fact, and was checked by direct differentiation before being written in here.

A second real, genuinely new anchor for this exact sub-case surfaced during the lesson's own audit trail and is not visible anywhere in the lesson's current worked content, which uses $\int x^2e^x\,dx$ only: June 2022, WMA14/01, Q8(b), "Find $\int x^2e^{-x}\,dx$" (10-mark question, Question Paper Log Number P71381A). The real mark scheme's own bullets, verbatim: M1 — "Attempts integration by parts in the right direction" (7 words) to reach $x^2e^{-x}\ldots\int xe^{-x}\,dx$; dM1 — the same, one step further, to $x^2e^{-x}\ldots xe^{-x}\ldots\int e^{-x}\,dx$; plus an explicitly credited DI alternative, "look for first two rows of the table to have correct forms for M1". Worked against those exact two method marks:

$$u=x^2,\ \tfrac{dv}{dx}=e^{-x}\Rightarrow \tfrac{du}{dx}=2x,\ v=-e^{-x}:\quad \int x^2e^{-x}dx=-x^2e^{-x}+2\int xe^{-x}\,dx\quad[M1]$$ $$u=x,\ \tfrac{dv}{dx}=e^{-x}\Rightarrow \tfrac{du}{dx}=1,\ v=-e^{-x}:\quad \int xe^{-x}dx=-xe^{-x}-e^{-x}+c\quad[dM1]$$ $$\int x^2e^{-x}dx=-x^2e^{-x}+2(-xe^{-x}-e^{-x})+c=-e^{-x}(x^2+2x+2)+c\quad[A1]$$

Checked: $\frac{d}{dx}\left[-e^{-x}(x^2+2x+2)\right]=e^{-x}(x^2+2x+2)-e^{-x}(2x+2)=e^{-x}x^2$. ✓ The general reduction formula above reproduces this exactly with $k=-1$, an internal consistency check.

#2.4 Parts — repeated, cyclic

Spec-named: $\int e^x\sin x\,dx$, worked in full in the cited lesson, including the one VERIDIAN-original observation the lesson itself flags (the inconsistent-choice tautology $I=I$). Generalised, since the lesson only shows $a=b=1$: for real constants $a,b$ (not both zero),

$$\int e^{ax}\sin(bx)\,dx=\frac{e^{ax}\left(a\sin bx-b\cos bx\right)}{a^2+b^2}+c,\qquad \int e^{ax}\cos(bx)\,dx=\frac{e^{ax}\left(a\cos bx+b\sin bx\right)}{a^2+b^2}+c$$

Both checked by direct differentiation in general $a,b$ (not merely at the lesson's $a=b=1$ case): for the cosine form, differentiating and collecting gives $\frac{e^{ax}}{a^2+b^2}\left[(a^2+b^2)\cos bx\right]=e^{ax}\cos bx$, confirming it exactly; the sine form checks the same way. A short new micro-example with different coefficients, testing coefficient-tracking through the cycle in a way $a=b=1$ cannot (since there is no coefficient to track when both constants are 1):

$$\int e^{2x}\cos x\,dx,\quad\text{first: }u=\cos x,\ \tfrac{dv}{dx}=e^{2x}\Rightarrow I=\tfrac12 e^{2x}\cos x+\tfrac12\int e^{2x}\sin x\,dx$$ $$\text{second (same factor differentiated again): }\int e^{2x}\sin x\,dx=\tfrac12e^{2x}\sin x-\tfrac12 I$$ $$I=\tfrac12e^{2x}\cos x+\tfrac14e^{2x}\sin x-\tfrac14I\;\Rightarrow\;\tfrac54I=e^{2x}\!\left(\tfrac12\cos x+\tfrac14\sin x\right)\;\Rightarrow\;I=\tfrac{e^{2x}}{5}(2\cos x+\sin x)+c$$

which matches the general formula above at $a=2,b=1$ exactly, and checks by differentiation to $e^{2x}\cos x$. (VERIDIAN-original; not from any past paper — included because it makes the cyclic mechanism generalise visibly beyond the one pair of coefficients the spec happens to name.)

#2.5 Substitution — self-devised

Spec-named: $\int x\sqrt{x-2}\,dx$, worked in full in the cited lesson, both indefinite and as the definite integral $\int_3^6 x\sqrt{x-2}\,dx$ with limits converted. One real structural nuance the lesson's own example can't show, because its substitution $u=x-2$ has $\frac{du}{dx}=+1$: when the substitution's derivative is negative, converting the limits reverses their order, and that reversal has to be tracked, not treated as an error. Short new example:

$$\int_0^1 x(1-x)^3\,dx,\quad u=1-x\;\Rightarrow\;x=1-u,\ \frac{du}{dx}=-1,\ dx=-du$$

When $x=0$, $u=1$; when $x=1$, $u=0$ — the limits swap order because $u$ decreases as $x$ increases.

$$\int_{x=0}^{1}x(1-x)^3dx=\int_{u=1}^{0}(1-u)u^3(-du)=\int_{u=0}^{1}(1-u)u^3\,du$$

(the minus sign from $dx=-du$ and the reversal of the limit order cancel each other — dropping either one alone, but not both, gives the wrong sign)

$$=\int_0^1(u^3-u^4)\,du=\left[\frac{u^4}{4}-\frac{u^5}{5}\right]_0^1=\frac14-\frac15=\frac1{20}$$

Cross-checked directly by expanding the original integrand: $x(1-x)^3=x-3x^2+3x^3-x^4$, integrating term by term over $[0,1]$ gives $\frac12-1+\frac34-\frac15=\frac1{20}$ exactly, confirming the substitution route. (VERIDIAN-original; not from any past paper.) This is a genuinely distinct failure surface from the one the facts bank documents for this paper (Oct 2022, Q7(i): "use of the x limits...instead of the u limits" — never converting at all); this sub-case is about a direction error that survives even when both limits are converted, if the reversal itself is missed.

#2.6 Substitution — given

The spec states plainly, and briefly: "the substitution will be given in more complicated integrals" (facts bank §1, 10 words). No second worked example accompanies this sentence anywhere in the spec's own guidance column, and the facts bank does not record one — this is stated here as a genuine gap in the evidence, not filled with an invented "typical" example. What can be said honestly: the cited lesson's own chain-drill block already exercises this exact sub-case ($\int x(2x-1)^4\,dx$, substitution $u=2x-1$ handed to the student rather than devised), so it is not undocumented in this course, only undocumented by the spec itself beyond the one sentence.

#2.7 Method choice as its own examined skill

Not a sub-case of parts or substitution, but a real, separately-tested skill that sits above both: knowing when neither applies because [[integration by recognition]] (spec 5.2, WMA13) already solves the integral in one line, or when parts is reached for out of habit on an integral a different technique handles more efficiently. The one confirmed instance in the facts bank sits formally under a different spec point (6.1, volume of revolution, not 6.2 itself — flagged as such deliberately, matching the cited lesson's own series-count discipline): Oct 2022, Q5, where examiners recorded candidates "wasted much time pursuing incorrect methods which included...integration by parts" (12 words) on an integral whose efficient route was recognition or substitution. It is included here, clearly labelled as adjacent rather than 6.2-native evidence, because it is the only real, sourced instance in this facts bank of integration by parts being named as the wrong tool, which is itself a real exam angle on this technique that a same-spec-point-only reading would miss entirely.

#Part 3 — Every real exam angle this topic has been tested from

Six categories, each with its own citation; nothing below is inferred beyond what its source states.

1. Mechanical slip inside a single application — differentiating or integrating the chosen factor incorrectly partway through an otherwise correct method. Oct 2022, Q7(ii): "a common error was then to integrate/differentiate the term cos2x incorrectly" (11 words), during a second parts application.
2. Coefficient lost across a multi-step derivation — a constant produced by one step of working not surviving to every term it should reach. Oct 2022, Q7(ii), a separate finding in the same question: "some candidates failed to realise they required a factor of 3" (11 words). Independently, Oct 2021, Q6, on the substitution side: "losing the factor 4 in the expression" (7 words) — the same failure mode, confirmed on both techniques independently, not just one.
3. An unjustified algebraic shortcut breaking the marked chain of reasoning — Oct 2021, Q8(b): "many candidates assumed that ln(x²) was identical to 2 ln x" (11 words) mid-derivation; the identity is true, the failure documented is using it as an invisible jump a line-by-line mark scheme can't follow.
4. A credited alternative method, confirmed rather than assumed — Jan 2024, Q5(a): "a few used the tabular 'DI' approach" (7 words), against the majority who used the traditional method — both scored on equal footing. Independently reconfirmed by the June 2022 mark scheme's own explicit DI-method note for Q8(b) (§2.3 above). Two independent primary-source confirmations for this specific credited alternative, out of the sources reviewed for this paper.
5. A conceptual substitution error, not an arithmetic one — Oct 2021, Q6: "candidates who changed sin 2x to sin"$^{-1}\!\left(\frac{u-3}{4}\right)$ — applying an inverse function to part of a substitution rather than substituting directly. Confirmed once, in one series checked.
6. Definite-integral limit handling — two distinct, separately confirmed failure shapes in the same question, Oct 2022, Q7(i): never converting the limits at all ("use of the x limits...instead of the u limits", 9 words) and the coefficient-loss failure already listed under category 2. This document's own §2.5 adds a third, undocumented-in-the-facts-bank shape (limit-order reversal) as an original, checked illustration, not a claimed exam finding.

No likelihood, frequency, or grade-boundary figure is stated anywhere above beyond "confirmed once, in one series checked" or "two independent primary-source confirmations, out of the sources reviewed" — the evidence base for this exact spec point is three examiner-report series with a 6.2-specific finding (Oct 2021, Oct 2022, Jan 2024, out of four full reports reviewed for this paper; January 2021's report contributed none) plus one mark-scheme-only anchor (June 2022, Q8b), and nothing beyond that count is claimed.

#Part 4 — What this document does not claim

The evidence base for spec 6.2 specifically is narrow relative to some other WMA14 spec points reviewed in the facts bank (contrast the 6+ series documented for proof by contradiction, spec 1.1): three examiner-report series and one additional mark-scheme anchor. Where this document generalises beyond that evidence — the two closed-form reduction formulas in §2.3–2.4, and the two new micro-examples in §2.2 and §2.5 — it says so at the point it appears, and none of that generalised content is presented as an exam-board fact. No second real, spec-cited example of a given substitution (§2.6) was found in the facts bank; that gap is stated rather than filled. No grade boundary, mark-loss percentage, or "commonly seen" frequency claim appears anywhere in this document that isn't a direct count of series checked.

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Not affiliated with or endorsed by Pearson Edexcel. Every prose fragment quoted above from a specification, mark scheme or examiner's report is under fifteen words with its own series/question attribution, per this course's copyright discipline; all formulae are transcribed as mathematical content, not prose. Sources: research/veridian/WMA14-verified-facts.md (§1 specification table, §4 general marking guidance, §5.6 topic findings, and the lesson-audit-log entry for integration-by-parts-and-substitution.ts dated 2026-09-14, which supplied the June 2022 Q8(b) anchor); content/veridian/curriculum/wma14/integration-by-parts-and-substitution.ts (full read, cross-referenced throughout rather than duplicated); content/veridian/technique-hub/wma14/codex-formula-and-method-bank.md (checked to confirm this document does not restate its Part 2 §2.7 method skeleton). The two general reduction formulas in Part 2 and the three new micro-examples (§2.2, §2.4, §2.5) are this document's own original synthesis, each checked by direct differentiation before being written in — not sourced from any Pearson document, and labelled as such at the point each appears.