12 (a) It is given that y = (ln 3x) / x² for x > 0. Find dy/dx. Give your answer in the form (A+B ln 3x) / x³, where A and B are integers. (b) Hence find ∫(ln 3x) / x³ dx.

Mark scheme

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

1. [4] B1 (for 12a) for 1/x or 3/3x
2. [4] M1 for a correct attempt at a quotient or a product
3. [4] A1 for all terms apart from log derivative correct.
4. [4] A1 (1-2ln3x)/x³
5. [4] M1 (for 12b) For using integration as a reverse of differentiation (reverse of part (a)) Allow using their A and B but do not allow any extra terms added or subtracted from their part (a)
6. [4] B1 FT on their A
7. [4] M1 dep for rearranging simplification and an attempt to integrate ax⁻³. Allow if +c is missing
8. [4] A1 must include +c ((-1 - 2ln3x)/(4x²) + c or -1/(4x²) - (ln3x)/(2x²) + c oe)