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WMA11/01 Oct 2022, Question 2: Integration: Undoing Differentiation

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 7 marks

The question, in short

Given a cubic identity with constants a and b, find a and b, then use the result to integrate the product divided by 5√x, giving each term in simplest form.

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

Part (a) · 2 marks

Find the constants a and b by comparing coefficients in the given cubic identity.

  1. 1(x − 5)(2x + 1) = 2x² − 9x − 5
  2. 2(2x² − 9x − 5)(x + 3) = 2x³ − 3x² − 32x − 15
  3. 3a = 2, b = −3

Answera = 2, b = −3

Part (b) · 5 marks

Hence integrate the product (x − 5)(2x + 1)(x + 3) divided by 5√x, writing each term in simplest form.

  1. 1(2x³ − 3x² − 32x − 15)/(5√x) = (2/5)x^(5/2) − (3/5)x^(3/2) − (32/5)x^(1/2) − 3x^(−1/2)
  2. 2∫(2/5)x^(5/2) dx = (2/5)(2/7)x^(7/2) = (4/35)x^(7/2)
  3. 3∫(3/5)x^(3/2) dx = (3/5)(2/5)x^(5/2) = (6/25)x^(5/2)
  4. 4∫(32/5)x^(1/2) dx = (32/5)(2/3)x^(3/2) = (64/15)x^(3/2)
  5. 5∫3x^(−1/2) dx = 6x^(1/2)
  6. 6Answer: (4/35)x^(7/2) − (6/25)x^(5/2) − (64/15)x^(3/2) − 6x^(1/2) + c

Answer(4/35)x^(7/2) − (6/25)x^(5/2) − (64/15)x^(3/2) − 6x^(1/2) + c

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