AQA AS Level Mathematics Paper 2, June 2025: Question 9

✅ Correct Answer

16

📐 Step-by-Step Working

1. Factor out the constant multiple from the integrand:
   ∫₁³ 2f(x) dx = 2 × ∫₁³ f(x) dx
2. Substitute the given value ∫₁³ f(x) dx = 8 :
   2 × 8 = 16

💡 Key Knowledge

Constant Multiple Rule: For any real constant k , ∫ₐᵇ k f(x) dx = k ∫ₐᵇ f(x) dx . Geometrically, this represents a vertical stretch of scale factor k , multiplying the bounded area by k .

❌ Common Errors

• Attempting to square the 2 or integrating 2 as 2x and multiplying it incorrectly.
• Changing the limits of integration when applying a vertical stretch.

✅ Correct Answer

12

📐 Step-by-Step Working

1. Split the integral across the addition:
   ∫₁³ (f(x) + 2) dx = ∫₁³ f(x) dx + ∫₁³ 2 dx
2. Substitute the known value of the first integral:
   = 8 + ∫₁³ 2 dx
3. Integrate the constant term 2 over limits 1 to 3 :
   ∫₁³ 2 dx = [2x]₁³ = 2(3) - 2(1) = 6 - 2 = 4
   (Alternatively: rectangle of height 2 and width (3 - 1) = 2 × 2 = 4)
4. Combine the two values:
   8 + 4 = 12

🧠 Exam Technique: Geometric Visualisation

The transformation y = f(x) + 2 translates the curve 2 units upwards. The region between the curve and the x-axis now has an extra rectangle beneath it with base width (3 - 1) = 2 and height 2 . Total area = 8 + (2 × 2) = 12 .

❌ Common Errors

• The "+2" Trap: Adding 2 directly to the integral answer: 8 + 2 = 10 . You must integrate the constant 2 with respect to x !
• Using the wrong width of the interval (e.g., calculating 2 × 3 = 6 instead of 2 × (3 - 1) = 4 ).

✅ Correct Answer

8

📐 Step-by-Step Working

1. Method 1: Transformation of Graphs
   The function y = f(x - 1) represents a translation of y = f(x) by the vector [+1, 0]ᵀ (1 unit to the right).
   The original interval [1, 3] shifted 1 unit right becomes [1 + 1, 3 + 1] = [2, 4] .
   Since horizontal translation preserves area, the value is identical: 8.
2. Method 2: Substitution (Formal)
   Let u = x - 1 ⇒ du = dx .
   When x = 2 , u = 2 - 1 = 1 .
   When x = 4 , u = 4 - 1 = 3 .
   Therefore: ∫₂⁴ f(x - 1) dx = ∫₁³ f(u) du = 8 .

💡 Key Knowledge

Definite integrals depend entirely on the shape of the function and the relative range of integration. Translating the function and its boundaries by the exact same shift does not change the integral's value:

∫ₐ₊ₖᵇ⁺ᵏ f(x - k) dx = ∫ₐᵇ f(x) dx

❌ Common Errors

• Thinking f(x - 1) translates 1 unit left and expecting the limits to be [0, 2] .
• Subtracting 1 from the integral value: 8 - 1 = 7 .
• Overcomplicating the question by attempting to find an algebraic expression for f(x) . Notice the command word is "Write down"—meaning 1 mark and no complex algebra required!