5 A curve $C$ is defined parametrically by

$$x = \frac { 2 } { \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } } \quad \text { and } \quad y = \frac { \mathrm { e } ^ { t } - \mathrm { e } ^ { - t } } { \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } }$$

for $0 \leqslant t \leqslant 1$. The area of the surface generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis is denoted by $S$.\\
(i) Show that $S = 4 \pi \int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { t } - \mathrm { e } ^ { - t } } { \left( \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } \right) ^ { 2 } } \mathrm {~d} t$.\\

(ii) Using the substitution $u = \mathrm { e } ^ { t } + \mathrm { e } ^ { - t }$, or otherwise, find $S$ in terms of $\pi$ and e .\\

\hfill \mbox{\textit{CAIE FP1 2019 Q5}}