O-LevelAdditional MathematicsDifferentiation and integration4037/12 · May/June 2022 · Q78 Marks

Show that 2/(2x+3) - 1/(x-1) + 1/(x-1)² can be written as (8-3x)/((x-1)²(2x+3)).

(a)Show that 2/(2x+3) - 1/(x-1) + 1/(x-1)² can be written as (8-3x)/((x-1)²(2x+3)).[2]
(b)Find ∫ [a to 2] (8-3x)/((x-1)²(2x+3)) dx where a > 2. Give your answers in the form c + ln d, where c and d are functions of a.[6]

Mark scheme

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

[M1] Attempt at a fraction, allow with an extra (x-1) term in each term of the numerator and the denominator
[A1] AG – must see sufficient detail to justify the given result, if an extra (x-1) term involved, it must be dealt with correctly
[1] B1 for ln|2x+3|
[1] B1 for -ln|x-1|
[1] B1 for -1/(x-1)
[M1] Dep on at least one ln term from integration, for applying limits correctly in their integral
[1] A1 for (a-2)/(a-1) or 1 - 1/(a-1)
[1] A1 for ln((2a+3)/(7(a-1))) oe

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 8 marks.
Source: O-Level Additional Mathematics, May/June 2022 Paper 1 Variant 2, question 7.

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