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WMA11/01 Oct 2024, Question 3: Integration: Undoing Differentiation

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

Given f(x) = (x + 5)²/√x for x > 0, integrate it, then use the derivative to show a quadratic equation and find the x where the derivative is zero.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 4 marks

Find the indefinite integral of (x + 5)²/√x, showing all stages of working.

  1. 1Expand and divide by √x: f(x) = x^(3/2) + 10x^(1/2) + 25x^(−1/2)
  2. 2∫ x^(3/2) dx = (2/5)x^(5/2), ∫ 10x^(1/2) dx = (20/3)x^(3/2), ∫ 25x^(−1/2) dx = 50x^(1/2)
  3. 3∫ f(x) dx = (2/5)x^(5/2) + (20/3)x^(3/2) + 50x^(1/2) + c

Answer(2/5)x^(5/2) + (20/3)x^(3/2) + 50x^(1/2) + c

Part (bi) · 4 marks

Differentiate f(x) and show that setting the derivative to zero gives 3x² + 10x − 25 = 0.

  1. 1f(x) = x^(3/2) + 10x^(1/2) + 25x^(−1/2)
  2. 2f′(x) = (3/2)x^(1/2) + 5x^(−1/2) − (25/2)x^(−3/2)
  3. 3Set f′(x) = 0 and multiply by x^(3/2): (3/2)x² + 5x − 25/2 = 0
  4. 4Multiply by 2: 3x² + 10x − 25 = 0

AnswerShown: f′(x) = (3/2)x^(1/2) + 5x^(−1/2) − (25/2)x^(−3/2) = 0; × 2x^(3/2) ⇒ 3x² + 10x − 25 = 0

Part (bii) · 1 mark

Using the quadratic from part (i), state the value of x that makes f′(x) = 0.

  1. 13x² + 10x − 25 = (3x − 5)(x + 5) = 0
  2. 2x = 5/3 or x = −5; since x > 0, x = 5/3 only

Answerx = 5/3

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Also searched as: WMA11 October 2024 · WMA11/01 · wma11 oct24 · IAS Maths Pure Mathematics 1 October 2024 · question 3.
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