Technique Masterclasses

Pure Mathematics 1

Deep Dive: Splitting Fractions Before Integrating

CODEX Deep-Dive: Splitting an Algebraic Fraction Before Integrating (WMA11, spec 5.1 / 5.2)

Provenance (2026-09-29). Built from a direct read of research/veridian/WMA11-verified-facts.md — §1 (spec table of contents, verbatim spec 5.1/5.2 wording and the memorised integration formula); §4.5 Integration (the Jan 2023 Q3 entry and its 2026-09-14 mark-scheme-bullet-coverage-audit update, itself a fresh pdftotext -layout/-raw re-extraction of the primary Jan 2023 QP/MS/ER, not a trust of any earlier paraphrase); §6 and §7 (series-coverage honesty notes); and the full "Lesson-audit log" entry for splitting-fractions-before-integrating.ts (2026-09-14). Also read in full: the existing lesson content/veridian/curriculum/wma11/splitting-fractions-before-integrating.ts (to confirm what it already teaches and does not duplicate its worked-chain/marked-solution content, which is deliberately VERIDIAN-original, not the real Q3); content/veridian/technique-hub/wma11/signal-topic-brief-splitting-fractions-before-integrating.md (the existing frequency/context synthesis); codex-formula-and-method-bank.md §2.5.1 (the compressed procedure, checked for overlap); forge-exam-technique-guide.md (its distinct Jan 2023 Q7(c)–(d) shifted-denominator finding, cross-referenced in §3.4 below); and content/veridian/question-bank/wma11.ts, grepped and confirmed to contain no entry tagged to this lesson. content/veridian/specCoverage.ts was also checked and contains no WMA11 entries at all (it covers other papers only), so no additional spec-guidance text exists for this point beyond §1's own transcription. 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 full worked chain, chain-drill and marked solution on this skill, deliberately built with VERIDIAN-original numbers so the lesson can combine spec 5.2 (splitting) with spec 5.1 (finding c from a point) in one item — something the one real verified anchor, Jan 2023 Q3, does not itself test. The SIGNAL brief already states the honest frequency count (1 of 5 series fully mined) and the mark-tariff comparison against the sibling two-constant topic. The formula bank already gives the four-line compressed procedure and its one governing citation.

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

  1. The real Jan 2023 Q3 numbers, worked through in full — the actual fraction, the actual split, and the actual final answer, none of which appears in the lesson itself (which uses different, original numbers).
  2. A general, provable argument, independent of any specific numbers, for why "integrate the numerator, integrate the denominator, then divide" can never produce a correct antiderivative for a term of this shape — generalising the lesson's own two numeric counterexamples into a single algebraic fact.
  3. A worked account of exactly which index each of the three real mark-scheme-named "wrong splits" gets correct, explaining mechanically why each still banks the integration method mark under the real M1 criterion, rather than just stating that they do.
  4. An explicit boundary of scope — the one spec-consistent denominator shape (a root) for which no confirmed real WMA11 instance was found, and the one confirmed real WMA11 error (a shifted denominator) that looks like this skill but is not — stated as honest gaps and boundaries, not invented exam angles.

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

1. Where this sits in the spec, precisely

Two spec points compose this skill. 5.1 requires "a constant of integration" every time indefinite integration reverses a differentiation — every time, not only when a question happens to supply enough information to find its numeric value. 5.2 gives the actual rule, but only for one shape of expression: xⁿ, and sums, differences or constant multiples of it, "excluding n = −1." The formula itself is one of the handful the spec states must be memorised rather than looked up: ∫xⁿ dx = xⁿ⁺¹/(n+1) + c, n ≠ −1.

That rule is stated for a single power of x. A fraction with more than one term on top and one term on the bottom — (Ax^m + Bx^k)/(Cx^n) — is not, as written, a power of x at all; it is one algebraic object sitting on the page, and 5.2's rule has nothing in it that mentions division. The move this topic teaches exists entirely to close that gap: rewrite the fraction, before any integration happens, as a sum of terms that genuinely are powers of x, using two facts from elsewhere in the spec rather than anything new — 1.1's index law xᵃ/xᵇ = xᵃ⁻ᵇ, and the ordinary arithmetic fact that (A+B)/D = A/D + B/D (adding two fractions that already share a denominator, run backwards). Once every term is a bare xⁿ, 5.2's rule has something to act on, term by term.

2. The general mechanism: why splitting first is not optional

2.1 The rewrite itself

For a numerator with two terms over a single monomial denominator, the split is mechanical: divide each numerator term by the denominator separately, then apply the index law to each resulting quotient.

Ax^m/(Cx^n) + Bx^k/(Cx^n) = (A/C)x^(m−n) + (B/C)x^(k−n)

Nothing about this step is specific to integration — it is the same identity that lets 6/x² + 3/x² be added into 9/x², done in reverse, followed by one application of the index law to each piece. The identity generalises to any number of numerator terms without any new idea (A/D + B/D + E/D = (A+B+E)/D run backwards, term by term), though no real WMA11 instance with three or more numerator terms has been found in the facts bank — the one confirmed anchor (§3.1 below) has exactly two.

2.2 A general proof that "integrate top, integrate bottom, divide" cannot work

The lesson's own worked counter-example shows two specific fractions (4x/x and x²/x) where integrating the numerator and denominator separately and then dividing gives the wrong function. That is enough to disprove the method, but it leaves open whether the two examples chosen happened to be unlucky. They did not — the failure is structural, and provable for any term of this shape.

Take a single term Ax^m over Cx^n (m ≠ n, and neither exponent equal to −1, so every integral below is a valid application of 5.2's rule). The wrong method integrates numerator and denominator as if they were unrelated expressions, then divides:

∫Ax^m dx / ∫Cx^n dx = [A/(m+1)·x^(m+1)] / [C/(n+1)·x^(n+1)] = [A(n+1)/(C(m+1))] · x^(m−n)

The correct method splits first, then integrates the single resulting power of x:

∫[(A/C)x^(m−n)] dx = A/(C(m−n+1)) · x^(m−n+1)

These two expressions do not merely have different coefficients — they carry different powers of x: x^(m−n) against x^(m−n+1), for every choice of A, C, m and n. The wrong method's result is systematically one power of x lower than any genuine antiderivative of the term, because it has already "used up" the +1 that integration is supposed to add, on the numerator and denominator separately, before the division that should have happened first ever takes place. This is why the lesson's simplest counter-example (4x/x) can look deceptively close to correct: the fraction 4x/x itself simplifies to the constant 4, and the wrong method's result is also 4 — but the required answer to ∫(4x/x) dx is 4x + c, a degree-1 expression, not the bare degree-0 constant the wrong method stops at. The coincidence is that h(x) itself equals the wrong method's output, not that the wrong method has integrated anything.

3. Real and spec-consistent sub-cases, each with a short micro-example

3.1 Integer-power monomial denominator — the one confirmed real instance

Jan 2023 Q3 (WMA11/01, Question Paper Log Number P72066A; mark scheme WMA11_01_MS_2301), verbatim: "Find ∫(4x⁵+3)/(2x²) dx, giving your answer in simplest form." Worked through in full — a derivation that does not appear anywhere else in this repository, since the lesson itself uses different, original numbers:

4x⁵/(2x²) = 2x³, and 3/(2x²) = (3/2)x⁻². So the integrand splits to 2x³ + (3/2)x⁻². Neither exponent is −1, so 5.2's rule applies to both: ∫2x³ dx = (1/2)x⁴, and ∫(3/2)x⁻² dx = (3/2)·x⁻¹/(−1) = −(3/2)x⁻¹. The finished answer is (1/2)x⁴ − (3/2)x⁻¹ + c — independently recomputed here and matching the audit-log's stated real mark-scheme answer exactly.

3.2 A root (fractional-index) denominator — spec-consistent, not confirmed in the real record

The formula bank names a monomial-root denominator, "3√x," as one example shape this method covers (§2.5.1), alongside the integer-power case above. No confirmed real WMA11 past-paper integration question using a root denominator was found anywhere in the facts bank while building this document — this is a genuine gap in the record, not a claim that such a question cannot occur. A short illustrative micro-example, constructed for this document rather than drawn from any paper: for (x² − 4)/(3√x) = (x² − 4)/(3x^(1/2)), the split gives (1/3)x^(3/2) − (4/3)x^(−1/2), and integrating gives (2/15)x^(5/2) − (8/3)x^(1/2) + c. The only new mechanical demand this shape adds over §3.1 is subtracting a fractional index (1/2) rather than an integer one — the split and integration rules themselves are unchanged.

3.3 The n = −1 exclusion — a real spec boundary, not yet confirmed as a tested trap

Spec 5.2 states the exclusion of n = −1 directly, and it is a genuine, checkable boundary: a split that lands a term on x⁻¹ cannot be integrated by 5.2's rule at all, because ∫x⁻¹ dx = ln|x| + c needs logarithms, which are P2 content. This can happen inside an otherwise ordinary split whenever a numerator's constant term sits over a denominator of exactly x¹ — for example, splitting (6x − 3)/x gives 6 − 3x⁻¹, where the second term is exactly the excluded case. No confirmed real WMA11 past-paper instance of a genuine question landing on this boundary was found in the facts bank; the lesson's own worked distractor using this exact shape (6x − 3)/x is VERIDIAN-original, built to match the spec's own stated exclusion, not a reproduction of a real question. The point is real and spec-mandated even though no live-exam instance of it has yet turned up in the series checked.

3.4 Boundary of scope: a shifted denominator is not this technique at all

A distinct, real, confirmed WMA11 error sits close enough to this skill that it is worth naming as a boundary rather than leaving a reader to over-generalise into it. Jan 2023 Q7(c)–(d) (a different question, on a line meeting a curve, spec strand 1.6, not this integration topic) drew an examiner-report finding, verbatim in part: "was often given as (6/x) – 3" — i.e. 6/(x − 2) mis-split into 6/x minus a constant. That is not a valid algebraic identity at all: unlike Ax^m/(Cx^n), a denominator with a shift inside it (x − a) cannot be separated additively, because dividing by (x − a) is not the same operation as dividing by x and then correcting by a constant. A shifted denominator can only be handled by cross-multiplying or genuine algebraic manipulation of the whole expression. This finding belongs to a different spec point and a different question family — it is reported here, following forge-exam-technique-guide.md's own cross-reference, only so that this document's own scope (a fixed monomial denominator under a multi-term numerator) is not silently over-applied to a shape it does not cover.

4. The real mark-dependency architecture, and why the three named wrong splits still bank a mark

Jan 2023 Q3's real mark scheme runs the 5 marks as M1 A1 (the split), then dM1 A1 A1 (the integration and simplification). Two structural facts, both re-extracted from the primary PDF rather than assumed: the split earns its own attempt/accuracy pair, not one combined mark — M1 for an attempt with one processed index correct, and a separate A1 for the fully correct split; and the integration mark is explicitly dM1, "depends on the first M mark," never an independent method mark of its own.

The real mark scheme also names three specific wrong splits that still bank the integration dM1: 8x³+6x⁻², 4x⁵+6x⁻², and 2x⁵+(3/2)x⁻² — because that mark checks that integration was correctly attempted on whatever split was written, not that the split itself was right. Comparing each against the real correct split (2x³ + (3/2)x⁻²) shows precisely why each one still clears the M1 bar of "one processed index correct": 8x³+6x⁻² has both indices right (3 and −2) with only the coefficients wrong, so either term alone satisfies the M1; 4x⁵+6x⁻² leaves the first term's index unprocessed (still 5, not reduced to 3) but the second term's index, −2, is exactly correct; 2x⁵+(3/2)x⁻² likewise leaves the first index unprocessed but reproduces the second term, (3/2)x⁻², exactly. In every one of the three named cases, at least one term's exponent has genuinely been produced by a correct application of the index law, even though the split as a whole is not fully correct — which is exactly the condition the M1 wording requires and the dependent dM1 then inherits.

This is also why the dependency runs from the M1, not the A1: an M1-level attempt (something, however imperfect, shaped like a two-term split with at least one correct index) is enough for a subsequent integration attempt to be assessed on its own terms, but a mark that specifically certifies the split's full accuracy cannot be inherited by a step that never required that full accuracy in the first place.

5. Real exam-angle synthesis

The one confirmed real appearance of this exact skill — a multi-term numerator split over a single monomial denominator, then integrated, worth 5 marks on its own as a standalone question — is Jan 2023 Q3, out of 5 WMA11 series fully mined question-by-question in the facts bank (Jan 2019, Oct 2020, Oct 2021, Jan 2023, Jan 2025). 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 confirmed failure modes, each attributable to the one named series and question: not recognising the fraction as a sum of two terms at all, rather than producing a wrong split — the examiner report describes candidates who "had no idea how to write the expression as the sum of two terms"; a distinct, smaller-scale error of attempting to integrate the three original terms (numerator's two terms and the denominator) separately before ever dividing, which the mark scheme credits with zero marks, not partial credit, because it is a different, wrong calculation rather than a slower correct one; and, on the finished, correctly-integrated answer, two named notation losses — omitting the constant of integration, and, separately, "not simplifying fully" or "missing out 'x' but writing the index" (both short quotations from the re-extracted Jan 2023 examiner report), which cost the final accuracy mark even when the integration method itself was sound.

Of these, only the omitted constant of integration is independently confirmed across more than this one question — it recurs in Oct 2021 Q1, Oct 2020 Q9 and Jan 2025 Q6(b), none of which involve a fraction to split first, making it a corpus-wide integration trap rather than one specific to this topic. The remaining traps named above — the no-idea-how-to-split failure, the integrate-before-dividing failure, and the missing-x/under-simplified notation loss — are, on the current record, confirmed on this one question only.

6. What is honestly still unverified

Eight of the nine WMA11 series this facts bank has any contact with have not been checked for this specific sub-skill (only Jan 2023 has); their silence is an honest research gap, not a claim that the skill is absent from them. No real WMA11 instance with three or more numerator terms, a root denominator, or a split landing on the excluded n = −1 case has been found — §3.2 and §3.3 above are spec-consistent, algebraically sound illustrations, explicitly not attributed to any real paper. Whether a fixed-power monomial denominator is ever combined, in a real question, with a point-substitution step in the way the lesson's own original worked examples do (rather than as two separate real questions, Q3 and Q11(b)) has not been confirmed either way. 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