- The curve \(C\) has equation
$$y = \frac { 1 } { 2 } \ln ( \operatorname { coth } x ) , \quad x > 0$$
- Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosech } 2 x$$
The points \(A\) and \(B\) lie on \(C\).
The \(x\) coordinates of \(A\) and \(B\) are \(\ln 2\) and \(\ln 3\) respectively.
- Find the length of the arc \(A B\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.
(6)