- (a) Use the definitions of hyperbolic functions in terms of exponentials to prove that
$$\begin{gathered}
1 - \operatorname { sech } ^ { 2 } x \equiv \tanh ^ { 2 } x
I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 3 } \ln 2 } \tanh ^ { n } 3 x \mathrm {~d} x \quad n \in \mathbb { Z } \quad n \geqslant 0
\end{gathered}$$
(b) Show that
$$I _ { n } = I _ { n - 2 } - \frac { p ^ { n - 1 } } { 3 ( n - 1 ) } \quad n \geqslant 2$$
where \(p\) is a rational number to be determined.
(c) Hence determine the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 3 } \ln 2 } \tanh ^ { 5 } 3 x \mathrm {~d} x$$
giving your answer in the form \(a \ln b + c\) where \(a , b\) and \(c\) are rational numbers to be found.