The diagram shows part of the curve y = 3 + 4/(2x+1) and the straight line 3y = 2x+6. Find the area of the shaded region, giving your answer in exact form.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [M1] for equating the line and curve and obtaining a 3 term quadratic expression in terms of x. (2x+6)/3 = 3+4/(2x+1) => 4x² - 4x - 15 = 0
2. [A1] For x-coordinate of the point of intersection. x = 5/2
3. [M1] for attempt to integrate with one term correct ∫(3 + 4/(2x+1))dx = 3x + 2ln(2x+1)
4. [Dep M1] for using their x-coordinate of C in their integral. Must have a term in ln(2x+1). [3x+2ln(2x+1)] evaluated.
5. [A1] Must be exact but allow unsimplified (15/2 + 2ln6)
6. [M1] for attempt at the trapezium, must have at least one side correct. If using integration, the integral must be correct using their 5/2. Area of trapezium = 85/12
7. [A1] Shaded area = 2ln6 + 5/12 or ln36 + 5/12 or ln6² + 5/12