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

Josh includes a constant 'k' in his solution, but he incorrectly places it in front of the logarithm. The correct integration should involve the natural logarithm of the variable with a constant added at the end, not multiplied with the logarithm. Specifically, the constant should not be there or should be treated as following the logarithmic term.

Both Floella and Georgia's answers reflect the same integral result. Floella uses a logarithmic form involving an absolute value, while Georgia correctly includes the constant '+ c'. To clarify, the arbitrary constant can take any form, including a positive or negative value, meaning that both forms can represent the same family of functions. Specifically, since |A| can represent any real number, we can conclude that their solutions are equivalent: