Mixed trigonometric products

8

1. Find \(\int \sin \theta \cos ^ { n } \theta d \theta\), where \(n \neq - 1\).
   Let \(I _ { m , n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { m } \theta \cos ^ { n } \theta d \theta\).
2. Show that, for \(m \geqslant 2\) and \(n \geqslant 0\), $$I _ { m , n } = \frac { m - 1 } { m + n } I _ { m - 2 , n }$$
3. 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.
4. Using the results given in parts (b) and (c), find the exact value of \(I _ { 2,5 }\).
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9 Using the substitution \(u = \cos \theta\), or any other method, find \(\int \sin \theta \cos ^ { 2 } \theta d \theta\). It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$I _ { n } = \frac { n - 1 } { n + 2 } I _ { n - 2 }$$ Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 4 } \theta \cos ^ { 2 } \theta d \theta\).

5 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \sin ^ { 2 } x \mathrm {~d} x\), for \(n \geqslant 0\). By differentiating \(\cos ^ { n - 1 } x \sin ^ { 3 } x\) with respect to \(x\), prove that $$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 } \quad \text { for } n \geqslant 2$$ Hence find the exact value of \(I _ { 4 }\).