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WMA11/01 Jan 2021, Question 9: Integrating with Two Conditions

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 11 marks

The question, in short

Integrate a squared linear expression divided by 4√x for x > 0, then use derivative and point conditions to determine a cubic f(x).

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

Part (i) · 5 marks

Find the indefinite integral of (3x + 2)² divided by 4√x, x > 0, in simplest form.

  1. 1(3x + 2)² = 9x² + 12x + 4
  2. 2Divide by 4√x: (9/4)x³ᐟ² + 3x¹ᐟ² + x⁻¹ᐟ²
  3. 3Integrate: (9/4)(2/5)x⁵ᐟ² + 3(2/3)x³ᐟ² + 2x¹ᐟ²
  4. 4= (9/10)x⁵ᐟ² + 2x³ᐟ² + 2x¹ᐟ² + c

Answer(9/10)x⁵ᐟ² + 2x³ᐟ² + 2x¹ᐟ² + c

Part (ii) · 6 marks

Given f′(x) = x² + ax + b, y-intercept −8, point P(3, −2) on C, and gradient 2 at P, find f(x).

  1. 1f′(3) = 2: 9 + 3a + b = 2, so b = −7 − 3a
  2. 2f(x) = (1/3)x³ + (1/2)ax² + bx + c, and c = −8 from the y intercept
  3. 3f(3) = −2: 9 + (9/2)a + 3b − 8 = −2
  4. 4Substitute b = −7 − 3a: 4.5a − 21 − 9a + 1 = −2, so a = −4
  5. 5b = −7 + 12 = 5
  6. 6f(x) = (1/3)x³ − 2x² + 5x − 8

Answerf(x) = (1/3)x³ − 2x² + 5x − 8

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