8 A curve has equation \(y = 2 \sqrt { x - 1 }\), where \(x > 1\). The length of the arc of the curve between the points on the curve where \(x = 2\) and \(x = 9\) is denoted by \(s\).
- Show that \(s = \int _ { 2 } ^ { 9 } \sqrt { \frac { x } { x - 1 } } \mathrm {~d} x\).
- Show that \(\cosh ^ { - 1 } 3 = 2 \ln ( 1 + \sqrt { 2 } )\).
- Use the substitution \(x = \cosh ^ { 2 } \theta\) to show that
$$s = m \sqrt { 2 } + \ln ( 1 + \sqrt { 2 } )$$
where \(m\) is an integer.
[0pt]
[6 marks]
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