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A-LevelMathematicsIntegration9709/22 · Feb/Mar 2022 · Q58 Marks

Given that y = tan²x, show that dy/dx = 2 tan x + 2 tan³ x.

  1. (a)Given that y = tan²x, show that dy/dx = 2 tan x + 2 tan³ x.[2]
  2. (b)Find the exact value of ∫(from π/6 to π/3) (tan x + tan²x + tan³x) dx.[6]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2022). The mark or mark type is shown in brackets.

  1. [B1] Differentiate to obtain 2 tan x sec² x
  2. [B1] Use sec² x = 1 + tan² x to confirm 2 tan x + 2 tan³ x [AG – necessary detail needed]
  3. [M1] Attempt to use part (a) result to integrate tan x + tan³ x
  4. [A1] Obtain (1/2) tan² x
  5. [M1] Use relevant identity to integrate tan² x
  6. [A1] Obtain sec² x - 1 and hence tan x - x
  7. [M1] Use limits correctly for integrand of form k₁ tan² x + k₂ tan x + k₃x
  8. [A1] Obtain √3 - π/12 [Or exact equivalent]

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About this question

Topic
Integration, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source
A-Level Mathematics, Feb/Mar 2022 Paper 2 Variant 2, question 5.

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