AQA
Further Paper 1
2020
June
— Question 12
6 marks
Exam Board
AQA
Module
Further Paper 1 (Further Paper 1)
Year
2020
Session
June
Marks
6
Topic
Hyperbolic functions
12
Use the definition of the cosh function to prove that
$$\cosh ^ { - 1 } \left( \frac { x } { a } \right) = \ln \left( \frac { x + \sqrt { x ^ { 2 } - a ^ { 2 } } } { a } \right) \quad \text { for } a > 0$$
[6 marks]
12
The formulae booklet gives the integral of \(\frac { 1 } { \sqrt { x ^ { 2 } - a ^ { 2 } } }\) as
$$\cosh ^ { - 1 } \left( \frac { x } { a } \right) \text { or } \ln \left( x + \sqrt { x ^ { 2 } - a ^ { 2 } } \right) + c$$
Ronald says that this contradicts the result given in part (a).
Explain why Ronald is wrong.