Given that 2x = tany, show that dy/dx = 2 / (1+4x²).

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [B1] Obtain 2 = sec²y dy/dx or equivalent
2. [M1] Use sec²y = 1 + tan² y
3. [A1] Replace tany with 2x and rearrange to obtain given answer dy/dx = 2/(1+4x²)
4. [M1] Integrate by parts and reach ax² tan⁻¹ 2x + b ∫ x²/(1+4x²) dx
5. [A1] Obtain x² tan⁻¹ 2x - ∫ x²/(1+4x²) dx
6. [M1] Reduce integral to expression of the form ∫ (m + n/(1+4x²)) dx
7. [M1] Complete integration and reach px² tan⁻¹ 2x + qx + r tan⁻¹ 2x
8. [A1] Obtain (1/2)x² tan⁻¹ 2x - (1/4)x + (1/8)tan⁻¹ 2x
9. [DM1] Use limits of x = 1/2 and x=±√3 in the correct order, having integrated twice
10. [A1] Obtain answer (π/8) - (√3/8) + (1/8)√2