AQA A-Level Mathematics Paper 1, June 2025: Question 5

Mark scheme

Show the mark scheme

Q Marking instructions AO Marks Typical solution

5 Integrates at least one term 1

1.1a M1 −

correctly, may be unsimplified 4x − 3 + x 2 dx

Integrates at least two terms

1.1a M1

correctly, may be unsimplified 1

12 2

Obtains 2x2 − 3x + 2x2 + c 4x x

= −3x + + c

22 1

or 2x − 3x + 2 x + c 1.1b A1 2

2x2 − 3x + 2x0.5 + c 2

or = 2x − 3x + 2x 2 + c

Question 5 Total 3

How to answer it

Indefinite Integration with Fractional Indices

📌 What this question tests

Applying the power rule for integration: ∫ xⁿ dx = (xⁿ⁺¹)/(n + 1) + c (for n ≠ -1).
Integrating individual terms term-by-term (linear combination of polynomial and fractional powers).
Working accurately with negative fractional indices (-1/2 + 1 = 1/2) and division by a fraction.
Remembering the constant of integration (+ c) in an indefinite integral.

Question 5 • 3 Marks

Evaluate the Indefinite Integral

Find ∫ (4x - 3 + x⁻¹/²) dx

📐 Step-by-Step Solution

1 Integrate each term individually:

For 4x (which is 4x¹):
4(x¹⁺¹)/(1 + 1) = 4x²/2 = 2x²
For -3 (which is -3x⁰):
-3(x⁰⁺¹)/(0 + 1) = -3x
For x⁻¹/²:
Add 1 to the power: -1/2 + 1 = 1/2
Divide by the new power: x¹/² / (1/2) = 2x¹/²

2 Combine all integrated terms:
2x² - 3x + 2x¹/²

3 Add the constant of integration:
2x² - 3x + 2x¹/² + c

✅ Final Answer & Mark Breakdown

Accepted equivalent forms:

2x² - 3x + 2x¹/² + c
2x² - 3x + 2√x + c
2x² - 3x + 2x⁰·⁵ + c

Mark Scheme Breakdown:
• M1 (AO 1.1a): Integrates at least one term correctly (can be unsimplified, e.g. 4x²/2 or x¹/² / (1/2)).
• M1 (AO 1.1a): Integrates at least two terms correctly (can be unsimplified).
• A1 (AO 1.1b): Completely correct, fully simplified expression including + c.

💡 Key Knowledge

The Power Rule: Increase the power by 1, then divide by the new power:
∫ xⁿ dx = (xⁿ⁺¹)/(n + 1)
Dividing by Fractions: Dividing by 1/2 is identical to multiplying by 2:
x¹/² / (1/2) = 2x¹/² = 2√x
Indefinite Integrals: When there are no limits of integration, always include an arbitrary constant + c .

🧠 Exam Technique & Examiner Insight

Quick Verification: Differentiate your final answer to verify it matches the integrand:
d/dx (2x² - 3x + 2x¹/² + c) = 4x - 3 + x⁻¹/². Takes 5 seconds and guarantees 3/3!
Partial Marks are Generous: Even if you struggle with the fractional index, getting 2x² - 3x secures the first two method marks (M1 M1). Never leave an integration question blank!
Unsimplified forms get method marks: Writing down 4x²/2 - 3x immediately secures 2 marks before you even tackle the fraction.

❌ Common Pitfalls & Traps to Avoid

Forgetting the "+ c": The final accuracy mark (A1) requires a complete solution. Leaving out + c is an immediate loss of the final mark!
Differentiating instead of Integrating: Confusing the operations and subtracting 1 from the power (e.g. writing -1/2 x⁻³/²) is a very common slip under exam pressure.
Arithmetic errors with negative fractions: Incorrectly computing -1/2 + 1 = -3/2 or failing to invert the denominator (e.g. leaving it as (1/2)x¹/² instead of 2x¹/² ).
Dropping the constant term's x: Integrating a constant -3 gives -3x , not 0 (which is the derivative).

Topics

Pure Mathematics · H: Integration