A-Level · Mathematics · AQA · Mark scheme decoded

AQA A-Level Mathematics: Fundamental Theorem of Calculus — mark scheme explained

Machine-verifiedchecked against the AQA A-Level Mathematics specificationlast verified 2 July 2026

The short answer

The Fundamental Theorem of Calculus (FTC) is a cornerstone of calculus that links the concept of differentiation and integration. It consists of two parts, each providing a different perspective on how these operations are related.

The question

Evaluate ∫ 0 3 (2 x + 1) d x using Part 2 of the FTC. [Paraphrased for study — not reproduced from any exam paper.]

Mark scheme, decoded

What each mark is really for — in plain English — and the wording trap that loses it.

S1
Find an antiderivative F ( x ) of f ( x ) = 2 x + 1.
S2
F ( x ) = ∫ (2 x + 1) d x = x 2 + x + C.
S3
Apply Part 2 of the FTC: ∫ 0 3 (2 x + 1) d x = F (3) - F (0).
S4
F (3) = 3 2 + 3 = 9 + 3 = 12.
S5
F (0) = 0 2 + 0 = 0.
S6
Therefore, ∫ 0 3 (2 x + 1) d x = 12 - 0 = 12.

Model answer

Worked through, with each step tagged to the mark it earns.

S1
Find an antiderivative F ( x ) of f ( x ) = 2 x + 1.
S2
F ( x ) = ∫ (2 x + 1) d x = x 2 + x + C.
S3
Apply Part 2 of the FTC: ∫ 0 3 (2 x + 1) d x = F (3) - F (0).
S4
F (3) = 3 2 + 3 = 9 + 3 = 12.
S5
F (0) = 0 2 + 0 = 0.
S6
Therefore, ∫ 0 3 (2 x + 1) d x = 12 - 0 = 12.
Final answer: 12

Common mistakes

Forgetting to apply the limits of integration in Part 2 of the FTC. — Always remember to substitute the upper limit into the antiderivative, then subtract the value obtained by substituting the lower limit.
Confusing Part 1 and Part 2 of the FTC. — Review the statements of both parts of the FTC to understand their distinct applications.
Forgetting the constant of integration when finding an antiderivative. — Always include the constant of integration C when finding an antiderivative unless you are evaluating a definite integral.
Incorrectly applying the FTC to non-continuous functions. — Always verify that the function is continuous on the given interval before applying the FTC.
Forgetting to check if a function has an antiderivative. — Be aware of the types of functions that do and do not have elementary antiderivatives, and use appropriate methods or numerical integration when necessary.

Where the marks go

Full worked solution (all marking points)5 marks