8 Let $I _ { n } = \int _ { 0 } ^ { \ln 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n } \mathrm {~d} x$.\\
(i) Show that

$$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right) \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n - 1 } \right] = n \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n } - 4 ( n - 1 ) \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n - 2 } .$$

(ii) Hence show that

$$n I _ { n } = 4 ( n - 1 ) I _ { n - 2 } + \frac { 3 } { 2 } \left( \frac { 5 } { 2 } \right) ^ { n - 1 } .$$

(iii) Use the result in part (ii) to find the $y$-coordinate of the centroid of the region bounded by the axes, the line $x = \ln 2$ and the curve

$$y = \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 2 } .$$

Give your answer correct to 3 decimal places.

\hfill \mbox{\textit{CAIE FP1 2007 Q8 [10]}}