Reduction formula or recurrence

4.

1. Show that, for \(n \geqslant 2\)
2. Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that $$\int x ^ { 4 } \cos x \mathrm {~d} x = \mathrm { f } ( x ) \sin x + \mathrm { g } ( x ) \cos x + c$$ where \(c\) is an arbitrary constant. $$I _ { n } = \int x ^ { n } \cos x \mathrm {~d} x$$
   1. Show that, for \(n \geqslant 2\) $$I _ { n } = x ^ { n } \sin x + n x ^ { n - 1 } \cos x - n ( n - 1 ) I _ { n - 2 }$$
   2. Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that

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.

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 \(A\).

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 \(I _ { 2 }\).

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

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

5 Let $$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$ where \(n \geqslant 0\).

1. Show that, for all \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
2. Hence prove by induction that, for all positive integers \(n\), $$I _ { n } < n ! .$$

The integral \(I_n\) is defined by $$I_n = \int_0^{\frac{1}{4}\pi} \sec^n x \, dx.$$ By considering \(\frac{d}{dx}(\tan x \sec^n x)\), or otherwise, show that $$(n + 1)I_{n+2} = 2^{\frac{4n}{n}} + nI_n.$$ [4] Find the value of \(I_6\). [4]

Let $$I_n = \int_0^1 (1 + x^2)^{-n} dx,$$ where \(n \geqslant 1\). By considering \(\frac{d}{dx}(x(1 + x^2)^{-n})\), or otherwise, prove that $$2nI_{n+1} = (2n - 1)I_n + 2^{-n}.$$ [5] Deduce that \(I_3 = \frac{3}{32}\pi + \frac{1}{4}\). [3]

It is given that \(I_n = \int_{1}^{e} (\ln x)^n \mathrm{d}x\) for \(n \geqslant 0\). Show that $$I_n = (n - 1)[I_{n-2} - I_{n-1}] \text{ for } n \geqslant 2.$$ [6] Hence find, in an exact form, the mean value of \((\ln x)^3\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm{e}\). [6]

Let \(I_n = \int_0^1 (1+3x)^n e^{-3x} dx\), where \(n\) is an integer.

1. Show that \(3I_n = 1 - 4^n e^{-3} + 3nI_{n-1}\). [3]
2. Find the exact value of \(I_2\). [3]

$$I_n = \int \sec^n x \, dx \quad n \geq 0$$

1. Prove that for \(n \geq 2\) $$(n-1)I_n = \tan x \sec^{n-2} x + (n-2)I_{n-2}$$ [6]
2. Hence, showing each step of your working, find the exact value of $$\int_0^{\pi/4} \sec^6 x \, dx$$ [4]

$$I_n = \int_1^e x^2 (\ln x)^n dx, \quad n \geq 0$$

1. Prove that, for \(n \geq 1\), $$I_n = \frac{e^3}{3} - \frac{n}{3} I_{n-1}$$ [4]
2. Find the exact value of \(I_3\). [4]

$$I_n = \int_0^{\pi} x^n \sin x \, dx$$

1. Show that for \(n \geq 2\) $$I_n = n \left( \frac{\pi}{2} \right)^{n-1} - n(n-1)I_{n-2}$$ [4]
2. Hence obtain \(I_3\), giving your answers in terms of \(\pi\). [4]

(Total 8 marks)