20 The integral \(I _ { n }\) is defined by
$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } \cos ^ { n } x \mathrm {~d} x \quad ( n \geq 0 )$$
20
- Show that
$$I _ { n } = \left( \frac { n - 1 } { n } \right) I _ { n - 2 } + \frac { 1 } { n \left( 2 ^ { \frac { n } { 2 } } \right) } \quad ( n \geq 2 )$$
20
- Use the result from part (a) to show that
$$\int _ { 0 } ^ { \frac { \pi } { 4 } } \cos ^ { 6 } x d x = \frac { a \pi + b } { 192 }$$
where \(a\) and \(b\) are integers to be found.
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