Trigonometric power reduction

8. $$I _ { n } = \int \cos ^ { n } x \mathrm {~d} x \quad n \geqslant 0$$

1. Prove that for \(n \geqslant 2\) $$I _ { n } = \frac { 1 } { n } \cos ^ { n - 1 } x \sin x + \frac { n - 1 } { n } I _ { n - 2 }$$
2. Show that for positive even integers \(n\) $$\int _ { 0 } ^ { \overline { 2 } } \cos ^ { n } x d x = \frac { ( n - 1 ) ( n - 3 ) \ldots 5 \times 3 \times 1 } { n ( n - 2 ) ( n - 4 ) \ldots 6 \times 4 \times 2 } \times \overline { 2 }$$
3. Hence determine the exact value of $$\int _ { 0 } ^ { \overline { 2 } } \cos ^ { 6 } x \sin ^ { 2 } x d x$$

   GUV SIHI NI JIVM ION OC

   VJYV SIHI NI JIIIM ION OC

   VJ4V SIHIANI JIIIM ION OO

1. Given that

$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } \cos ^ { n } \theta \mathrm {~d} \theta , \quad n \geqslant 0$$

1. prove that, for \(n \geqslant 2\), $$n I _ { n } = \left( \frac { 1 } { \sqrt { 2 } } \right) ^ { n } + ( n - 1 ) I _ { n - 2 }$$
2. Hence find the exact value of \(I _ { 5 }\), showing each step of your working.

5. $$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x , \quad 0 < x < \frac { \pi } { 2 } , \quad n \geqslant 0$$

1. Show that, for \(n \geqslant 2\) $$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { 1 } { n - 1 } \cot x \operatorname { cosec } ^ { n - 2 } x$$
2. Hence, or otherwise, find $$\int \operatorname { cosec } ^ { 4 } x \mathrm {~d} x$$ giving your answer in terms of \(\cot x\).

5. $$\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { \mathrm { n } } x \mathrm { dx } , \mathrm { n } \geqslant 0$$

1. Show that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\), for \(n \geqslant 2\)
2. Using the result in part (a), find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \sin ^ { 5 } x \cos x d x$$

4. (i) $$f ( x ) = x \arccos x \quad - 1 \leqslant x \leqslant 1$$ Find the exact value of \(f ^ { \prime } ( 0.5 )\).
(ii) $$\mathrm { g } ( x ) = \arctan \left( \mathrm { e } ^ { 2 x } \right)$$ Show that $$\mathrm { g } ^ { \prime \prime } ( x ) = k \operatorname { sech } ( 2 x ) \tanh ( 2 x )$$ where \(k\) is a constant to be found. \includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-15_2647_1840_119_114}

1. Prove that for \(n \geqslant 2\) $$( n - 1 ) I _ { n } = \tan x \sec ^ { n - 2 } x + ( n - 2 ) I _ { n - 2 }$$
2. Hence, showing each step of your working, find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec ^ { 6 } x d x$$ $$I _ { n } = \int \sec ^ { n } x \mathrm {~d} x \quad n \geqslant 0$$ Prove that for \(n > 2\)

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

1. Prove that for \(n \geqslant 2\) $$I _ { n } = \frac { 1 } { n } \left( - \sin ^ { n - 1 } x \cos x + ( n - 1 ) I _ { n - 2 } \right)$$ Given that \(n\) is an odd number, \(n \geqslant 3\)
2. show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x = \frac { ( n - 1 ) ( n - 3 ) \ldots 6.4 .2 } { n ( n - 2 ) ( n - 4 ) \ldots 7.5 .3 }$$
3. Hence find \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 5 } x \cos ^ { 2 } x d x\)

9

1. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$ Show that, for \(n \geqslant 2\), $$n I _ { n } = ( n - 1 ) I _ { n - 2 } .$$
2. The equation of a curve, in polar coordinates, is $$r = \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \pi$$ (a) Find the equations of the tangents at the pole and sketch the curve.
   (b) Find the exact area of the region enclosed by the curve. RECOGNISING ACHIEVEMENT

4 It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \mathrm {~d} x\) for \(n \geqslant 0\).

1. Show that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geqslant 2\).
2. Hence find \(I _ { 11 }\) as an exact fraction.

9

1. Using the substitution \(u = \tan x\), or otherwise, find \(\int \sec ^ { 2 } x \tan ^ { 2 } x \mathrm {~d} x\).
   It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \tan ^ { 2 } x \mathrm {~d} x$$
2. Using the result that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x ) = \tan x \sec x\), show that, for \(n \geqslant 2\), $$( n + 1 ) I _ { n } = ( \sqrt { } 2 ) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 }$$
3. Hence find the mean value of \(\sec ^ { 4 } x \tan ^ { 2 } x\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\), giving your answer in exact form.

8 Given that \(I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x \quad n \geq 0\) show that \(n I _ { n } = ( n - 1 ) I _ { n - 2 } \quad n \geq 2\) [0pt] [5 marks]

7 Let \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { \mathrm { n } } \mathrm { xdx }\) for integers \(n \geqslant 0\).

1. Show that, for \(n \geqslant 2 , \quad \mathrm { nl } _ { \mathrm { n } } = ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } - 2 }\).
2. Use this reduction formula to deduce the exact value of \(I _ { 8 }\).
3. Use the results of parts (a) and (b) to determine the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 6 } x \sin ^ { 2 } x d x\).

5 In this question you must show detailed reasoning.
It is given that \(I _ { n } = \int _ { 0 } ^ { \pi } \sin ^ { n } \theta \mathrm {~d} \theta\) for \(n \geq 0\).

1. Prove that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geq 2\).
2. (a) Evaluate \(I _ { 1 }\).
   (b) Use the reduction formula to determine the exact value of \(\int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta \sin ^ { 5 } \theta \mathrm {~d} \theta\).

5. $$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x \quad n \in \mathbb { Z }$$

1. Prove that, for \(n \geqslant 2\) $$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { \operatorname { cosec } ^ { n - 2 } x \cot x } { n - 1 }$$
2. Hence show that $$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \operatorname { cosec } ^ { 6 } x \mathrm {~d} x = \frac { 56 } { 135 } \sqrt { 3 }$$

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

1. Prove that for \(n \geqslant 2\) $$I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$$
2. Hence determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } 64 \sin ^ { 5 } x \cos ^ { 5 } x d x$$

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

1. Prove that, for \(n \geqslant 2\), $$n I _ { n } = ( n - 1 ) I _ { n - 2 }$$
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-22_588_1018_630_520} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} A designer is asked to produce a poster to completely cover the curved surface area of a solid cylinder which has diameter 1 m and height 0.7 m . He uses a large sheet of paper with height 0.7 m and width of \(\pi \mathrm { m }\).
   Figure 2 shows the first stage of the design, where the poster is divided into two sections by a curve. The curve is given by the equation $$y = \sin ^ { 2 } ( 4 x ) - \sin ^ { 10 } ( 4 x )$$ relative to axes taken along the bottom and left hand edge of the paper.
   The region of the poster below the curve is shaded and the region above the curve remains unshaded, as shown in Figure 2. Find the exact area of the poster which is shaded.

5 In this question you must show detailed reasoning.
It is given that \(I _ { n } = \int _ { 0 } ^ { \pi } \sin ^ { n } \theta \mathrm {~d} \theta\) for \(n \geq 0\).

1. Prove that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geq 2\).
2. Evaluate \(I _ { 1 }\) and use the reduction formula to determine the exact value of \(\int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta \sin ^ { 5 } \theta \mathrm {~d} \theta\).

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

1. 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
2. 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. \includegraphics[max width=\textwidth, alt={}]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-31_2491_1755_173_123} number \section*{Additional page, if required.
   Additional page, if required. Write the question numbers in the left-hand margin.} Additional page, if required. number Additional page, if required.
   Write the question numbers in the left-hand margin. number \section*{Additional page, if required.
   Additional page, if required. Write the question numbers in the left-hand margin.} number\section*{Additional page, if required.
   Additional page, if required. Write the question numbers in the left-hand margin.}
   \includegraphics[max width=\textwidth, alt={}]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-36_2487_1748_175_130}