Polynomial times trigonometric

6. $$I _ { n } = \int _ { 0 } ^ { \sqrt { \frac { \pi } { 2 } } } x ^ { n } \cos \left( x ^ { 2 } \right) \mathrm { d } x \quad n \geqslant 1$$

1. Prove that, for \(n \geqslant 5\) $$I _ { n } = \frac { 1 } { 2 } \left( \frac { \pi } { 2 } \right) ^ { \frac { n - 1 } { 2 } } - \frac { 1 } { 4 } ( n - 1 ) ( n - 3 ) I _ { n - 4 }$$
2. Hence, determine the exact value of \(I _ { 5 }\), giving your answer in its simplest form.

4. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } x ^ { n } \sin 2 x \mathrm {~d} x , \quad n \geqslant 0$$

1. Prove that, for \(n \geqslant 2\), $$I _ { n } = \frac { 1 } { 4 } n \left( \frac { \pi } { 4 } \right) ^ { n - 1 } - \frac { 1 } { 4 } n ( n - 1 ) I _ { n - 2 }$$
2. Find the exact value of \(I _ { 2 }\)
3. Show that \(I _ { 4 } = \frac { 1 } { 64 } \left( \pi ^ { 3 } - 24 \pi + 48 \right)\)

6. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } x ^ { n } \sin x \mathrm {~d} x$$

1. Show that for \(n \geq 2\) $$I _ { n } = n \left( \frac { \pi } { 2 } \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
2. Hence obtain \(I _ { 3 }\), giving your answers in terms of \(\pi\).

5 It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \cos x \mathrm {~d} x$$

1. Prove that, for \(n \geqslant 2\), $$I _ { n } = \left( \frac { 1 } { 2 } \pi \right) ^ { n } - n ( n - 1 ) I _ { n - 2 } .$$
2. Find \(I _ { 4 }\) in terms of \(\pi\).

6 It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \pi } x ^ { n } \sin x \mathrm {~d} x$$

1. Prove that, for \(n \geqslant 2 , I _ { n } = \pi ^ { n } - n ( n - 1 ) I _ { n - 2 }\).
2. Find \(I _ { 5 }\) in terms of \(\pi\).

8

1. Given that $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } t ^ { n } \sin t \mathrm {~d} t$$ show that, for \(n \geqslant 2\), $$I _ { n } = n \left( \frac { \pi } { 2 } \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 } .$$
2. A curve \(C\) in the \(x - y\) plane is defined parametrically in terms of \(t\). It is given that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = t ^ { 4 } ( 1 - \cos 2 t ) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = t ^ { 4 } \sin 2 t .$$ Find the length of the arc of \(C\) from the point where \(t = 0\) to the point where \(t = \frac { 1 } { 2 } \pi\).

7 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x\), where \(n\) is a non-negative integer. Show that $$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 } , \quad \text { for } n \geqslant 2$$ Find the exact value of \(I _ { 4 }\).

6 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x\).

1. Prove that, for \(n \geqslant 2\), $$I _ { n } + n ( n - 1 ) I _ { n - 2 } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } .$$
2. Calculate the exact value of \(I _ { 1 }\) and deduce the exact value of \(I _ { 3 }\).

9 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\). Given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x\), prove that, for \(n > 1\), $$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$ By first using the substitution \(x = \cos ^ { - 1 } u\), find the value of $$\int _ { 0 } ^ { 1 } \left( \cos ^ { - 1 } u \right) ^ { 3 } \mathrm {~d} u$$ giving your answer in an exact form.