A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Integration of Basic Functions and Trigonometric Functions — mark scheme explained
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
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For integration questions, marks are typically awarded for correctly applying the integration rules and including the constant of integration C . Partial credit may be given for correct steps even if the final answer is incorrect.
What the command words demand
- Integrate
- Find the indefinite integral of a given function.
- Evaluate
- Calculate the definite integral of a function over a specified interval.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Aim to spend about 2-3 minutes per mark on integration questions. Ensure you have enough time to check your work and include all necessary components, such as the constant of integration.
- Integrate each term separately.3 marks∫ (2x 4 ) dx = (2/5) x 5∫ (-3x 2 ) dx = -x 3∫ 1 dx = x
- Combine the results and add the constant of integration.1 mark(2/5) x 5 - x 3 + x + C
Final answer: (2/5) x 5 - x 3 + x + C
Work through every step correctly and you earn all 4 marks.
Another worked example
Find the indefinite integral of e -x + sin(2x) .
- Integrate each term separately.3 marks∫ e -x dx = -e -x∫ sin(2x) dx = -(1/2) cos(2x)
- Combine the results and add the constant of integration.1 mark-e -x - (1/2) cos(2x) + C
Final answer: -e -x - (1/2) cos(2x) + C
Work through every step correctly and you earn all 4 marks.
Common mistakes
Forgetting to add the constant of integration C .
Why it happens: Students often forget that the indefinite integral represents a family of functions, and the constant C accounts for this.
Fix: Always include + C at the end of your answer when finding an indefinite integral.
Incorrectly applying the power rule to x -1 .
Why it happens: The power rule does not apply when the exponent is -1. The integral of 1/x is ln|x| .
Fix: 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.
Why it happens: Students sometimes forget that the integral of e kx is (1/k) e kx . The coefficient k in the exponent affects the result.
Fix: Always divide by the coefficient of x when integrating exponential functions with a linear argument.
Incorrectly applying the integral of trigonometric functions.
Why it happens: Students may mix up the integrals of sine and cosine, or forget to include the coefficient in the denominator.
Fix: 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 .
Why it happens: The natural logarithm function, ln(x) , is only defined for positive values of x . The integral of 1/x must include the absolute value to cover both positive and negative values of x .
Fix: 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
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Indefinite Integration | Integrate a polynomial term by term and include the constant of integration. | 4 |
| Indefinite Integration | Integrate a sum of exponential and trigonometric terms, including the constant of integration. | 4 |
| Indefinite Integration | Integrate a trigonometric and reciprocal expression, remembering to include the constant of integration. | 4 |
| Indefinite Integration | Integrate an exponential and power term, remembering to add the constant of integration. | 4 |
| Total across these question types | 16 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.