Integration is Pure Topic 8, and it typically carries around 30 of the 300 marks across the two Pure-heavy papers — an estimate, since the exact split moves a little year to year. AS content (reversing differentiation for xⁿ, definite integrals and area under a curve) is examinable from Year 1; the A2 content — standard integrals of eᵏˣ, trig functions and 1/x, area between curves, substitution, integration by parts, partial fractions and separable differential equations — is added on top for the full A level.
With differentiation (Topic 7), integration forms the calculus core of the specification and feeds into kinematics, numerical methods and modelling questions elsewhere on the paper. Few integration questions stand alone — most also draw on algebraic manipulation, trig identities or partial fractions from earlier topics.
The specification statements this topic covers. AS = Year-1 content, also assessed in the standalone AS course (8MA0); A2 = full A level only. Typical share of a 300-mark series: ≈32 marks — our estimate from the 2018–2025 papers, not an official weighting.
| Ref | Spec statement | Level |
|---|---|---|
| 8.1 | Fundamental Theorem of Calculus | AS |
| 8.2 | Integrating xⁿ | AS |
| 8.2 | Standard integrals | A2 |
| 8.3 | Definite integrals and area | AS |
| 8.3 | Area between curves | A2 |
| 8.4 | Integration as limit of a sum | A2 |
| 8.5 | Substitution and by parts | A2 |
| 8.6 | Integration by partial fractions | A2 |
| 8.7 | Separable differential equations | A2 |
| 8.8 | Interpreting DE solutions | A2 |
Where a question asks for an exact value — common in definite integrals involving e, ln or a surd — a rounded decimal forfeits the final mark however accurate it is. Examiners repeatedly flag this across Pure topics, and it recurs in integration because so many standard results are ln or exponential terms.
In derivations ending in a printed answer, examiners report candidates jumping straight to the result or skipping a step such as setting an expression to zero before solving. Once a result is printed it must then be used for the rest of the question, even if your own working differs slightly.
Integrals of sin kx and cos kx are radian results, and examiners repeatedly flag calculator-mode confusion where evaluation quietly switches to degrees. Check the mode before evaluating any definite integral with a trig term.
A correct method can still lose its accuracy mark to a slip under pressure — a sign error through integration by parts, or a coefficient miscopied when substituting limits. Write each line out in full once negatives and fractions appear.
An indefinite integral is a family of curves, not one answer — omitting +c is an incomplete integration, and in 'find the curve given a point' or differential-equation questions it means the given condition was never used. Write +c the moment you integrate.
Integrating e^(kx), sin kx, cos kx or (ax+b)ⁿ reverses differentiation, so the linear coefficient inside needs dividing out (or the power divided by n+1) — not multiplying. If unsure, differentiate your answer and check it returns the original integrand.
The power rule has exactly one exponent it doesn't cover: x⁻¹. Applying it gives x⁰ over zero, which is meaningless — the actual result, ln|x| + c, is expected from memory.
A definite integral over a region below the x-axis returns a negative value, but area is always positive — and one integral spanning a crossing point can let the parts cancel. Sketch first, find the crossing points, and integrate each region separately.
Substituting u for a function of x changes the integrand, the dx, and the limits all at once — it's common to update only the first and leave the original x-limits in place. Convert the limits with the substitution, or convert the answer back to x.
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Know what the booklet gives and what it doesn't. It supplies the integration by parts formula and the integrals of sec²kx, tan kx, cot kx, cosec kx and sec kx, but ∫xⁿ dx and the integrals of cos kx, sin kx, e^(kx) and 1/x are not in the booklet — they need to be secure from memory because most of the topic builds on them.
Every paper allows a calculator, so the risk is misusing it rather than lacking one: check degree-versus-radian mode before evaluating a trig integral numerically, and use it to sanity-check an answer rather than to shortcut an exact-value or 'show that' question, where the algebraic working is what's actually marked.
Build the AS content — the reverse power rule, definite integrals and area under a curve — to fluency first, since substitution, parts, partial fractions and differential equations all lean on it. Revise partial fractions alongside their use here rather than separately, and practise writing out full working even when the answer is obvious — that's what 'show that' and exact-value marks check for.
All 19 topics: Edexcel A level Maths topic guides. Reference: formula booklet vs memorise and grade boundaries.
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Original questions written for the Pearson Edexcel A Level Mathematics (9MA0) specification. Not affiliated with or endorsed by Pearson.