8.
$$I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0$$
- Show that, for \(n \geqslant 1\)
- Hence show that
$$\int _ { 0 } ^ { \ln 2 } \tanh ^ { 4 } x \mathrm {~d} x = p + \ln 2$$
where \(p\) is a rational number to be found.
8. \(\quad I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0\) - Show that, for \(n \geqslant 1\)
$$I _ { n } = I _ { n - 1 } - \frac { 1 } { 2 n - 1 } \left( \frac { 3 } { 5 } \right) ^ { 2 n - 1 }$$