Four students, Tom, Josh, Floella and Georgia are attempting to complete the indefinite integral \[ \int \frac{1}{x} \: dx \quad \text{for } x > 0 \] Each of the students' solutions is shown below: Tom \[ \int \frac{1}{x} \: dx = \ln x \]\n Josh \[ \int \frac{1}{x} \: dx = k \ln x \]\n Floella \[ \int \frac{1}{x} \: dx = \ln A x \]\n Georgia \[ \int \frac{1}{x} \: dx = \ln x + c \] 6 (a) (i) Explain what is wrong with Tom's answer - AQA - A-Level Maths Mechanics - Question 6 - 2020 - Paper 1

Josh's answer includes a constant 'k' multiplied by ( \ln x ), which is incorrect because it suggests a dependence on an arbitrary constant that does not meet the requirements of the indefinite integral. The constant should not be there as a coefficient. The correct representation is [ \int \frac{1}{x} : dx = \ln x + c ].

Floella's answer ( \int \frac{1}{x} : dx = \ln A x ) can be rewritten as ( \ln A + \ln x ), exploiting the property of logarithms ( \ln(ab) = \ln a + \ln b ). This means that her answer can be interpreted as including a constant from ( \ln A ), which is equivalent to the constant 'c' in Georgia's answer. Thus, both answers are equivalent representations of the same integral.