Express \(17 \cosh x - 15 \sinh x\) in the form \(\mathrm { e } ^ { - \mathrm { x } } \left( \mathrm { ae } ^ { \mathrm { bx } } + \mathrm { c } \right)\) where \(a , b\) and \(c\) are integers to be determined.
A function is defined by \(\mathrm { f } ( x ) = \frac { 1 } { \sqrt { 17 \cosh x - 15 \sinh x } }\). The region bounded by the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \ln 3\) is rotated by \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
\section*{(b) In this question you must show detailed reasoning.}
Use a suitable substitution, together with known results from the formula book, to show that the volume of \(S\) is given by \(k \pi \tan ^ { - 1 } q\) where \(k\) and \(q\) are rational numbers to be determined.