Find \(\int \sin \theta \cos ^ { n } \theta d \theta\), where \(n \neq - 1\).
Let \(I _ { m , n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { m } \theta \cos ^ { n } \theta d \theta\).
Show that, for \(m \geqslant 2\) and \(n \geqslant 0\),
$$I _ { m , n } = \frac { m - 1 } { m + n } I _ { m - 2 , n }$$
By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 5 }\), where \(z = \cos \theta + i \sin \theta\), use de Moivre's theorem to show that
$$\cos ^ { 5 } \theta = a \cos 5 \theta + b \cos 3 \theta + c \cos \theta$$
where \(a\), \(b\) and \(c\) are constants to be determined.
Using the results given in parts (b) and (c), find the exact value of \(I _ { 2,5 }\).
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