8.06a Reduction formulae: establish, use, and evaluate recursively

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 $$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \operatorname { cosec } ^ { 6 } x \mathrm {~d} x = \frac { 56 } { 135 } \sqrt { 3 }$$

7. $$I _ { n } = \int \left( 4 - x ^ { 2 } \right) ^ { - n } \mathrm {~d} x \quad n > 0$$

1. Show that, for \(n > 0\) $$I _ { n + 1 } = \frac { x } { 8 n \left( 4 - x ^ { 2 } \right) ^ { n } } + \frac { 2 n - 1 } { 8 n } I _ { n }$$
2. Find \(I _ { 2 }\)

1. In this question you must show all stages of your working.

You must not use the integration facility on your calculator. $$I _ { n } = \int t ^ { n } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t \quad n \geqslant 0$$

1. Show that, for \(n > 1\) $$I _ { n } = \frac { t ^ { n - 1 } } { 5 ( n + 2 ) } \left( 4 + 5 t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - \frac { 4 ( n - 1 ) } { 5 ( n + 2 ) } I _ { n - 2 }$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1241b133-4161-4c04-9b50-067904cc25c2-20_385_394_829_833} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} The curve shown in Figure 1 is defined by the parametric equations $$x = \frac { 1 } { \sqrt { 5 } } t ^ { 5 } \quad y = \frac { 1 } { 2 } t ^ { 4 } \quad 0 \leqslant t \leqslant 1$$ This curve is rotated through \(2 \pi\) radians about the \(x\)-axis to form a hollow open shell.
2. Show that the external surface area of the shell is given by $$\pi \int _ { 0 } ^ { 1 } t ^ { 7 } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t$$ Using the results in parts (a) and (b) and making each step of your working clear,
3. determine the value of the external surface area of the shell, giving your answer to 3 significant figures.

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 $$\int _ { 0 } ^ { \frac { \pi } { 2 } } 64 \sin ^ { 5 } x \cos ^ { 5 } x d x$$

8. $$I _ { n } = \int _ { 0 } ^ { 2 } ( x - 2 ) ^ { n } \mathrm { e } ^ { 4 x } \mathrm {~d} x \quad n \geqslant 0$$

1. Prove that for \(n \geqslant 1\) $$I _ { n } = - a ^ { n - 2 } - \frac { n } { 4 } I _ { n - 1 }$$ where \(a\) is a constant to be determined.
2. Hence determine the exact value of $$\int _ { 0 } ^ { 2 } ( x - 2 ) ^ { 2 } e ^ { 4 x } d x$$

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$I _ { n } = \int \frac { \cos ( n x ) } { \sin x } \mathrm {~d} x \quad n \geqslant 1$$

1. Show that, for \(n \geqslant 1\) $$I _ { n + 2 } = \frac { 2 \cos ( n + 1 ) x } { n + 1 } + I _ { n }$$
2. Hence determine the exact value of $$\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \frac { \cos ( 5 x ) } { \sin x } d x$$ giving the answer in the form \(a + b \ln c\) where \(a , b\) and \(c\) are rational numbers to be found.

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.

6 In this question you must show detailed reasoning. It is given that \(I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } t ^ { n } \sqrt { 1 + t ^ { 2 } } \mathrm {~d} t\) for integers \(n \geqslant 0\).

1. Show that \(I _ { 1 } = \frac { 7 } { 3 }\).
2. Prove that, for \(n \geqslant 2 , ( n + 2 ) I _ { n } = 8 ( \sqrt { 3 } ) ^ { n - 1 } - ( n - 1 ) I _ { n - 2 }\). The curve \(C\) is defined parametrically by $$x = 10 t ^ { 3 } , y = 15 t ^ { 2 } \text { for } 0 \leqslant t \leqslant \sqrt { 3 }$$ When the curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
3. Determine
   • the values of the integers \(k\) and \(m\) such that \(A = k \pi I _ { m }\),
   • the exact value of \(A\).

6 For positive integers \(n\), the integrals \(I _ { n }\) are given by \(I _ { n } = \int _ { 1 } ^ { 5 } x ^ { n } \sqrt { 2 + x ^ { 2 } } \mathrm {~d} x\).

1. Show that \(I _ { 1 } = 26 \sqrt { 3 }\).
2. Prove that, for \(n \geqslant 3 , ( n + 2 ) I _ { n } = 3 \sqrt { 3 } \left( 27 \times 5 ^ { n - 1 } - 1 \right) - 2 ( n - 1 ) I _ { n - 2 }\).
3. Determine the exact value of \(I _ { 5 }\) as a rational multiple of \(\sqrt { 3 }\).

4. \(I _ { n } = \int _ { 0 } ^ { a } ( a - x ) ^ { n } \cos x \mathrm {~d} x , \quad a > 0 , \quad n \geqslant 0\)

1. Show that, for \(n \geqslant 2\), $$I _ { n } = n a ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
2. Hence evaluate \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( \frac { \pi } { 2 } - x \right) ^ { 2 } \cos x d x\)

5. $$I _ { n } = \int _ { 0 } ^ { 5 } \frac { x ^ { n } } { \sqrt { } \left( 25 - x ^ { 2 } \right) } \mathrm { d } x , \quad n \geq 0$$

1. Find an expression for \(\int \frac { x } { \sqrt { \left( 25 - x ^ { 2 } \right) } } \mathrm { d } x , \quad 0 \leq x \leq 5\).
2. Using your answer to part (a), or otherwise, show that $$I _ { n } = \frac { 25 ( n - 1 ) } { n } I _ { n - 2 } , \quad n \geq 2$$
3. Find \(I _ { 4 }\) in the form \(k \pi\), where \(k\) is a fraction.

9. $$I _ { n } = \int \left( x ^ { 2 } + 1 \right) ^ { - n } \mathrm {~d} x , \quad n > 0$$

1. Show that, for \(n > 0\), $$I _ { n + 1 } = \frac { x \left( x ^ { 2 } + 1 \right) ^ { - n } } { 2 n } + \frac { 2 n - 1 } { 2 n } I _ { n }$$
2. Find \(I _ { 2 }\).

6

1. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - x } x ^ { n } \mathrm {~d} x$$ Prove that, for \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
2. Evaluate \(I _ { 3 }\), giving the answer in terms of e.

12 The integral \(S _ { n }\) is defined by $$S _ { n } = \int _ { 0 } ^ { a } x ^ { n } \sinh x \mathrm {~d} x \quad ( n \geq 0 )$$ 12

1. Show that for \(n \geq 2\) $$S _ { n } = n ( n - 1 ) S _ { n - 2 } + a ^ { n } \cosh a - n a ^ { n - 1 } \sinh a$$

   12	Hence show that \(\int _ { 0 } ^ { 1 } x ^ { 4 } \sinh x d x = \frac { 9 } { 2 } e + \frac { 65 } { 2 } e ^ { - 1 } - 24\)

9 It is given that \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\) for \(n \geqslant 0\). Show that $$I _ { n } = ( n - 1 ) \left[ I _ { n - 2 } - I _ { n - 1 } \right] \text { for } n \geqslant 2$$ Hence find, in an exact form, the mean value of \(( \ln x ) ^ { 3 }\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm { e }\).

12

1. Let \(I _ { n } = \int \frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } } \mathrm {~d} x\), for integers \(n \geqslant 0\).
   By writing \(\frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } }\) as \(x ^ { n - 1 } \times \frac { x } { \sqrt { x ^ { 2 } + 1 } }\), or otherwise, show that, for \(n \geqslant 2\), $$n I _ { n } = x ^ { n - 1 } \sqrt { x ^ { 2 } + 1 } - ( n - 1 ) I _ { n - 2 } .$$
2. The diagram shows a sketch of the hyperbola \(H\) with equation \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 16 } = 1\). \includegraphics[max width=\textwidth, alt={}, center]{32ed7cc8-3456-4cf0-952a-ee04eada1298-6_593_666_776_776}
   1. Find the coordinates of the points where \(H\) crosses the \(x\)-axis.
   2. The curve \(J\) has parametric equations \(x = 2 \cosh \theta , y = 4 \sinh \theta\), for \(\theta \geqslant 0\). Show that these parametric equations satisfy the cartesian equation of \(H\), and indicate on a copy of the above diagram which part of \(H\) is \(J\).
   3. The arc of the curve \(J\) between the points where \(x = 2\) and \(x = 34\) is rotated once completely about the \(x\)-axis to form a surface of revolution with area \(S\). Show that $$S = 16 \pi \int _ { \alpha } ^ { \beta } \sinh \theta \sqrt { 5 \cosh ^ { 2 } \theta - 1 } \mathrm {~d} \theta$$ for suitable constants \(\alpha\) and \(\beta\).
   4. Use the substitution \(u ^ { 2 } = 5 \cosh ^ { 2 } \theta - 1\) to show that $$S = \frac { 8 \pi } { \sqrt { 5 } } ( 644 \sqrt { 5 } - \ln ( 9 + 4 \sqrt { 5 } ) )$$

12

1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 }$$
2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
   1. Sketch this curve.
   2. Find the exact length of the curve.

4 Let \(I _ { n } = \int _ { 0 } ^ { 4 } x ^ { n } \sqrt { 2 x + 1 } \mathrm {~d} x\) for \(n \geqslant 0\). Show that, for \(n \geqslant 1\), $$( 2 n + 3 ) I _ { n } = 27 \times 4 ^ { n } - n I _ { n - 1 }$$

12

1. (a) Show that \(\tanh x = \frac { \mathrm { e } ^ { 2 x } - 1 } { \mathrm { e } ^ { 2 x } + 1 }\).
   (b) Hence, or otherwise, show that, if \(\tanh x = \frac { 1 } { k }\) for \(k > 1\), then \(x = \frac { 1 } { 2 } \ln \left( \frac { k + 1 } { k - 1 } \right)\) and find an expression in terms of \(k\) for \(\sinh 2 x\).
2. A curve has equation \(y = \frac { 1 } { 2 } \ln ( \tanh x )\) for \(\alpha \leqslant x \leqslant \beta\), where \(\tanh \alpha = \frac { 1 } { 3 }\) and \(\tanh \beta = \frac { 1 } { 2 }\). Find, in its simplest exact form, the arc length of this curve.

12 The curve \(C\) has equation \(y = \ln \left( \tanh \frac { 1 } { 2 } x \right)\), for \(x > 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosech } x\).
2. For positive integers \(n\), the length of the arc of \(C\) between \(x = n\) and \(x = 2 n\) is \(L _ { n }\).
   1. Show by calculus that, when \(n\) is large, \(L _ { n } \approx n\).
   2. Explain how this result corresponds to the shape of \(C\).

12

1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 }$$
2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
   1. Sketch this curve.
   2. Find the exact length of the curve.

12

1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 } .$$
2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
   1. Sketch this curve.
   2. Find the exact length of the curve.

12

1. Let \(I _ { \mathrm { n } } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 }$$
2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
   1. Sketch this curve.
   2. Find the exact length of the curve.

12

1. The sequence \(\left\{ u _ { n } \right\}\) is defined for all integers \(n \geq 0\) by $$u _ { 0 } = 1 \quad \text { and } \quad u _ { n } = n u _ { n - 1 } + 1 , \quad n \geq 1 .$$ Prove by induction that \(u _ { n } = n ! \sum _ { r = 0 } ^ { n } \frac { 1 } { r ! }\).
2. (a) Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x\) for \(n \geq 0\), show that, for \(n \geq 1\), $$I _ { n } = n I _ { n - 1 } - \frac { 1 } { \mathrm { e } }$$ (b) Evaluate \(I _ { 0 }\) exactly and deduce the value of \(I _ { 1 }\).
   (c) Show that \(I _ { n } = n ! - \frac { u _ { n } } { \mathrm { e } }\) for all integers \(n \geq 1\).