A-Level · Mathematics · AQA · Mark scheme decoded

AQA A-Level Mathematics: Integration of Basic Functions and Trigonometric Functions — mark scheme explained

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

The short answer

In AQA A-Level Mathematics, integration is a fundamental concept that allows us to find the antiderivative (or indefinite integral) of various functions. This spec point focuses on integrating x n (excluding n = -1 ), and related sums, differences, and constant multiples.

The question

Find the indefinite integral of 2x 4 - 3x 2 + 1 . [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
Step 1: Integrate each term separately.
S2
∫ (2x 4 ) dx = (2/5) x 5
S3
∫ (-3x 2 ) dx = -x 3
S4
∫ 1 dx = x
S5
Step 2: Combine the results and add the constant of integration.
S6
(2/5) x 5 - x 3 + x + C

Model answer

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

S1
Step 1: Integrate each term separately.
S2
∫ (2x 4 ) dx = (2/5) x 5
S3
∫ (-3x 2 ) dx = -x 3
S4
∫ 1 dx = x
S5
Step 2: Combine the results and add the constant of integration.
S6
(2/5) x 5 - x 3 + x + C
Final answer: (2/5) x 5 - x 3 + x + C

Common mistakes

Forgetting to add the constant of integration C . — Always include + C at the end of your answer when finding an indefinite integral.
Incorrectly applying the power rule to x -1 . — Memorize that ∫ (1/x) dx = ln|x| + C and use this formula instead of the power rule.
Forgetting to divide by the coefficient of x when integrating exponential functions. — Always divide by the coefficient of x when integrating exponential functions with a linear argument.
Incorrectly applying the integral of trigonometric functions. — Memorize that ∫ sin(kx) dx = -(1/k) cos(kx) + C and ∫ cos(kx) dx = (1/k) sin(kx) + C . Always check the sign and the coefficient in the denominator.
Forgetting to use absolute value when integrating 1/x . — Always write ∫ (1/x) dx = ln|x| + C to ensure the logarithm is defined for all non-zero values of x .

Where the marks go

Full worked solution (all marking points)4 marks