4 It is given that the positive constant \(a\) is such that
$$\int _ { - a } ^ { a } \left( 4 \mathrm { e } ^ { 2 x } + 5 \right) \mathrm { d } x = 100$$
- Show that \(a = \frac { 1 } { 2 } \ln \left( 50 + \mathrm { e } ^ { - 2 a } - 5 a \right)\).
- Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 50 + \mathrm { e } ^ { - 2 a _ { n } } - 5 a _ { n } \right)\) to find \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
- Show that \(\frac { \cos 2 x + 9 \cos x + 5 } { \cos x + 4 } \equiv 2 \cos x + 1\).
- Hence find the exact value of \(\int _ { - \pi } ^ { \pi } \frac { \cos 4 x + 9 \cos 2 x + 5 } { \cos 2 x + 4 } \mathrm {~d} x\).