A-LevelMathematicsIntegration9709/31 · May/June 2022 · Q68 Marks

Let `I = ∫(from 0 to 3) (27 / (9 + x²)²) dx`.

(a)Using the substitution x = 3 tan θ, show that I = ∫(from 0 to π/4) cos² θ dθ.[4]
(b)Hence find the exact value of I.[4]

Mark scheme

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

[B1] State or imply dx = 3sec20d0
[M1] Substitute throughout for x and dx
[A1] Obtain any correct form in terms of 0 e.g. ∫ (81sec²θ / (9+9tan²θ)²) dθ
[A1] Justify change of limits and obtain ∫(from 0 to π/4) cos²θ dθ correctly AG
[*M1] Obtain indefinite integral of the form ∫ a + b cos 2θ dθ, where ab≠0
[A1] Obtain 1/2 θ + 1/4 sin 2θ
[DM1] Use correct limits correctly in an expression containing pθ and q sin 2θ where pq ≠ 0
[A1] Obtain answer 1/8(π+2) Or exact equivalent e.g. π/8 + 1/4

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.

Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source: A-Level Mathematics, May/June 2022 Paper 3 Variant 1, question 6.

Questions and mark schemes are from past examination papers and remain the copyright of the exam board. How Oracle Prep builds its content