A-LevelMathematicsIntegration9709/22 · Oct/Nov 2020 · Q610 Marks

Find ∫ (8/(4x+1) + 8/cos²(4x+1)) dx.

(a)Find ∫ (8/(4x+1) + 8/cos²(4x+1)) dx.[4]
(b)It is given that ∫[0 to π/4] (3 + 4 cos²(½x) + k sin 2x) dx = 10. Find the exact value of the constant k.[6]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

[1] Express 8/cos²(4x+1) as 8sec²(4x+1)
[1] Integrate to obtain the form aln(4x+1)
[1] Integrate to obtain btan(4x+1)
[1] Obtain 2ln|4x+1|+2tan(4x+1)+c Condone use of brackets rather than modulus signs
[1] Express 4cos²(1/2x) in the form p+qcosx For constants where pq ≠ 0
[1] Obtain correct 2+2cosx
[1] Integrate to obtain form px+qsinx+rcos2x For constants where pqr ≠ 0
[1] Obtain correct 5x+2sinx-k/2cos2x Allow 3x + 2x in place of 5x
[1] Apply limits correctly, equate to 10 and solve for k
[1] Obtain k = 8-5/2π CWO

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Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, Oct/Nov 2020 Paper 2 Variant 2, question 6.

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