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4PM1/01R Jun 2020, Question 6: Integrating eˣ, Trig and (ax+b)ⁿ

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 7 marks

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The question, in short

Prove the sum-to-product identity for sin(A+B)+sin(A−B), then apply it to rewrite 2sin7xcosx and evaluate a definite integral of 6sin7xcosx from 0 to π/4 using calculus.

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

Part (a) · 2 marks

Show that adding sin(A+B) and sin(A−B) gives 2sinAcosB.

  1. 1sin(A + B) = sin A cos B + cos A sin B
  2. 2sin(A − B) = sin A cos B − cos A sin B
  3. 3Adding gives 2sin A cos B

Answersin(A + B) + sin(A − B) = 2sin A cos B shown

Part (b) · 1 mark

Use part a to write 2sin7xcosx as sin mx + sin nx, stating integers m and n.

  1. 1A = 7x, B = x in (a)
  2. 22sin 7x cos x = sin 8x + sin 6x

Answersin 8x + sin 6x (m = 8, n = 6)

Part (c) · 4 marks

Using calculus, evaluate the integral from 0 to π/4 of 6sin7xcosx dx.

  1. 1∫ = 3∫ from 0 to π/4 of (sin 8x + sin 6x) dx
  2. 2= 3[−(1/8)cos 8x − (1/6)cos 6x] from 0 to π/4
  3. 3= 3[(−1/8 − 0) − (−1/8 − 1/6)]
  4. 4= 3 × 1/6 = 1/2

Answer1/2

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