Independent multi-part (different techniques)

8

1. Show that \(\int _ { 2 } ^ { 4 } 4 x \ln x \mathrm {~d} x = 56 \ln 2 - 12\).
2. Use the substitution \(u = \sin 4 x\) to find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 24 } \pi } \cos ^ { 3 } 4 x \mathrm {~d} x\).

1. (i) Find

$$\int \frac { 2 x ^ { 2 } + 5 x + 1 } { x ^ { 2 } } \mathrm {~d} x , \quad x > 0$$ (ii) Find $$\int x \cos 2 x \mathrm {~d} x$$

4. (i) Find \(\int \ln \left( \frac { x } { 2 } \right) \mathrm { d } x\).
(ii) Find the exact value of \(\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } x \mathrm {~d} x\).

6. (a) Find \(\int \tan ^ { 2 } x \mathrm {~d} x\).
(b) Use integration by parts to find \(\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x\).
(c) Use the substitution \(u = 1 + e ^ { x }\) to show that $$\int \frac { \mathrm { e } ^ { 3 x } } { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - \mathrm { e } ^ { x } + \ln \left( 1 + \mathrm { e } ^ { x } \right) + k$$ where \(k\) is a constant.

2. (i) Find $$\int x \cos \left( \frac { x } { 2 } \right) \mathrm { d } x$$ (ii) (a) Express \(\frac { 1 } { x ^ { 2 } ( 1 - 3 x ) }\) in partial fractions.
(b) Hence find, for \(0 < x < \frac { 1 } { 3 }\) $$\int \frac { 1 } { x ^ { 2 } ( 1 - 3 x ) } \mathrm { d } x$$

1. (i) Find

$$\int x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x$$ (4)
(ii) Use the substitution \(u = \sqrt { 1 - 3 x }\) to show that $$\int \frac { 27 x } { \sqrt { 1 - 3 x } } \mathrm {~d} x = - 2 ( 1 - 3 x ) ^ { \frac { 1 } { 2 } } ( A x + B ) + k$$ where \(A\) and \(B\) are integers to be found and \(k\) is an arbitrary constant.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Use the substitution \(u = \mathrm { e } ^ { x } - 3\) to show that $$\int _ { \ln 5 } ^ { \ln 7 } \frac { 4 \mathrm { e } ^ { 3 x } } { \mathrm { e } ^ { x } - 3 } \mathrm {~d} x = a + b \ln 2$$ where \(a\) and \(b\) are constants to be found.
2. Show, by integration, that $$\int 3 \mathrm { e } ^ { x } \cos 2 x \mathrm {~d} x = p \mathrm { e } ^ { x } \sin 2 x + q \mathrm { e } ^ { x } \cos 2 x + c$$ where \(p\) and \(q\) are constants to be found and \(c\) is an arbitrary constant.

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

\section*{Solutions based on calculator technology are not acceptable.}

1. Use integration by parts to find the exact value of $$\int _ { 0 } ^ { 4 } x ^ { 2 } \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ giving your answer in simplest form.
2. Use integration by substitution to show that $$\int _ { 3 } ^ { \frac { 21 } { 2 } } \frac { 4 x } { ( 2 x - 1 ) ^ { 2 } } \mathrm {~d} x = a + \ln b$$ where \(a\) and \(b\) are constants to be found.

6

1. Find \(\int x \cos 2 x d x\).
2. Using the substitution \(u = x ^ { 2 } + 1\), or otherwise, find the exact value of \(\int _ { 2 } ^ { 3 } \frac { x } { x ^ { 2 } + 1 } \mathrm {~d} x\).

3. Find

1. \(\int \frac { x } { 2 - x ^ { 2 } } \mathrm {~d} x\),
2. \(\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\).

1. (i) Find

$$\int x ^ { 2 } \sin x \mathrm {~d} x$$ (ii) Use the substitution \(u = 1 + \sin x\) to find the value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos x ( 1 + \sin x ) ^ { 3 } d x$$

8. (i) Find $$\int x ^ { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x$$ (ii) Using the substitution \(u = \sin t\), evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } 2 t \cos t \mathrm {~d} t$$

3 A circular ornamental garden pond, of radius 2 metres, has weed starting to grow and cover its surface. As the weed grows, it covers an area of \(A\) square metres. A simple model assumes that the weed grows so that the rate of increase of its area is proportional to \(A\). 3

1. Show that the area covered by the weed can be modelled by
   where \(B\) and \(k\) are constants and \(t\) is time in days since the weed was first noticed.
   [0pt] [4 marks] $$A = B \mathrm { e } ^ { k t }$$ 3
2. When it was first noticed, the weed covered an area of \(0.25 \mathrm {~m} ^ { 2 }\). Twenty days later the weed covered an area of \(0.5 \mathrm {~m} ^ { 2 }\) 3
3. 1. State the value of \(B\).
      [0pt] [1 mark] 3
4. (ii) Show that the model for the area covered by the weed can be written as $$A = 2 ^ { \frac { t } { 20 } - 2 }$$ [4 marks]
   Question 3 continues on the next page 3
5. (iii) How many days does it take for the weed to cover half of the surface of the pond?
   [0pt] [2 marks]
   3
6. State one limitation of the model.
   3
7. Suggest one refinement that could be made to improve the model. \(4 \quad \int _ { 1 } ^ { 2 } x ^ { 3 } \ln ( 2 x ) \mathrm { d } x\) can be written in the form \(p \ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\).
   [0pt] [5 marks]

4

1. Use integration by parts to find \(\int x \sin x \mathrm {~d} x\).
2. Using the substitution \(u = x ^ { 2 } + 5\), or otherwise, find \(\int x \sqrt { x ^ { 2 } + 5 } \mathrm {~d} x\).
3. The diagram shows the curve \(y = x ^ { 2 } - 9\) for \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-3_844_663_685_694} The shaded region \(R\) is bounded by the curve, the lines \(y = 1\) and \(y = 2\), and the \(y\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

5

1. Find \(\int \frac { 1 } { 3 + 2 x } \mathrm {~d} x\).
2. By using integration by parts, find \(\int x \sin \frac { x } { 2 } \mathrm {~d} x\).

3

1. Find \(\int \mathrm { e } ^ { 4 x } \mathrm {~d} x\).
2. Use integration by parts to find \(\int \mathrm { e } ^ { 4 x } ( 2 x + 1 ) \mathrm { d } x\).
3. By using the substitution \(u = 1 + \ln x\), or otherwise, find \(\int \frac { 1 + \ln x } { x } \mathrm {~d} x\).

8. (a) Integrate

1. \(\quad \mathrm { e } ^ { - 3 x + 5 }\)
2. \(x ^ { 2 } \ln x\) (b) Use an appropriate substitution to show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } } \mathrm {~d} x = \frac { \pi } { 12 } - \frac { \sqrt { 3 } } { 8 }$$

14. (i) Use integration by parts to find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x \mathrm {~d} x\).
(ii) Use the substitution \(x = \sin \theta\) to show that, for \(| x | \leq 1\), \(\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { x } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } + c\), where \(c\) is an arbitrary constant.

8

1. Using integration by parts, find \(\int x \sin ( 2 x - 1 ) \mathrm { d } x\).
2. Use the substitution \(u = 2 x - 1\) to find \(\int \frac { x ^ { 2 } } { 2 x - 1 } \mathrm {~d} x\), giving your answer in terms of \(x\).
   (6 marks)

6

1. Use integration by parts to find \(\int x \mathrm { e } ^ { 5 x } \mathrm {~d} x\).
2. 1. Use the substitution \(u = \sqrt { x }\) to show that $$\int \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x = \int \frac { 2 } { 1 + u } \mathrm {~d} u$$
   2. Find the exact value of \(\int _ { 1 } ^ { 9 } \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x\).

12 A random sample of 84 students was asked how many revision websites they had visited in the past month. The data is summarised in the table below. Find the interquartile range of the number of websites visited by these 84 students.
Circle your answer.
[0pt] [1 mark]
341942 Identify this Venn diagram. Tick ( ✓ ) one box. \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_506_501_584_374} \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_117_111_580_897} \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_508_504_580_1203} \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_117_120_580_1710} \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_508_501_1135_374} \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_112_111_1133_897} \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_505_506_1133_1201} \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_112_109_1133_1717} Turn over for the next question