← MarkMate · Blog

How to Mark A-level Integration by the Mark Scheme (+ the Marks Students Lose)

Integration is where careful students and careless ones separate — not on the calculus, but on the bookkeeping: the constant of integration, the limits, the sign at the end. Here's how examiners split the marks and where accuracy quietly leaks.

Indefinite integrals: the constant is a mark

For ∫ (6x² + 2) dx, a typical mark scheme reads:

• M1 — raising the power by one on at least one term (x^n → x^(n+1)/(n+1)). Awarded for the attempt.
• A1 — fully correct terms: 2x³ + 2x.
• B1 — including + c.

The + c is its own independent mark. Students who do perfect calculus and forget it lose a mark that has nothing to do with maths — pure habit.

Definite integrals: limits and subtraction

For a definite integral the marks usually run: M1 for integrating, A1 for the correct integrated expression, M1 for substituting the limits, A1 for the correct final value. The classic losses:

• Substituting limits the wrong way round — upper minus lower, not the reverse. A sign error here keeps the method marks but loses the final A1.
• Arithmetic on the substitution — the method is sound, the final value wrong → M1 yes, A1 no (often with follow-through on the next part).
• Leaving + c in a definite integral — it should cancel; carrying it shows a misunderstanding and can cost the accuracy mark.

Where students lose marks most

1. Forgetting + c

The single most common dropped mark in indefinite integration. It's a B mark — independent and unforgiving.

2. Power-rule slips on fractions and negatives

∫ x^(−2) dx trips students: the method mark holds, the accuracy mark goes on a sign or index slip.

3. Not simplifying when "exact" is required

A decimal where a fraction or surd was asked for can trigger cao and lose the final mark.

Marking it the examiner's way

Award the M for a valid integration attempt, the A only when every term (and the constant, where relevant) is exact, and treat the + c as its own check. For definite integrals, verify the limits were subtracted the right way round. Tell the student exactly which line lost the mark — "you integrated correctly but dropped the +c" teaches far more than a cross.