1.08i Integration by parts

6 \includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-08_351_1031_264_516} The diagram shows the curve \(\mathrm { y } = \mathrm { xe } ^ { - \mathrm { ax } }\), where \(a\) is a positive constant, and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 2 } { a } } x e ^ { - a x } d x\).

5

1. Show that \(\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 2 \cot 2 x\).
2. Hence solve the equation \(\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 4\), for \(0 < x < \pi\).

8 \includegraphics[max width=\textwidth, alt={}, center]{8c26235b-c78c-40d8-9e8e-213dc1311186-12_437_686_274_719} The diagram shows the curve \(y = x ^ { 3 } \ln x\), for \(x > 0\), and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { 1 } { 2 }\). [5]

10 \includegraphics[max width=\textwidth, alt={}, center]{5f80ae11-34c3-4d2f-89f8-71b4ac021c7d-16_426_908_262_616} The diagram shows the curve \(y = ( 2 - x ) \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the area of the shaded region bounded by the curve and the axes. Give your answer in terms of e.

10 \includegraphics[max width=\textwidth, alt={}, center]{19aff1b7-51b7-4d44-86e6-45dad32aa121-16_426_908_262_616} The diagram shows the curve \(y = ( 2 - x ) \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the area of the shaded region bounded by the curve and the axes. Give your answer in terms of e.

8 The constant \(a\) is such that \(\int _ { 1 } ^ { a } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x = 6\).

1. Show that \(a = \exp \left( \frac { 1 } { \sqrt { a } } + 2 \right)\). \(\left[ \exp ( x ) \right.\) is an alternative notation for \(\left. \mathrm { e } ^ { x } .\right]\)
2. Verify by calculation that \(a\) lies between 9 and 11 .
3. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. Using the expansions of \(\sin ( 3 x + 2 x )\) and \(\sin ( 3 x - 2 x )\), show that $$\frac { 1 } { 2 } ( \sin 5 x + \sin x ) \equiv \sin 3 x \cos 2 x$$
2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin 3 x \cos 2 x \mathrm {~d} x = \frac { 1 } { 5 } ( 3 - \sqrt { 2 } )\).

4 Find the exact value of \(\int _ { \frac { 1 } { 3 } \pi } ^ { \pi } x \sin \frac { 1 } { 2 } x \mathrm {~d} x\).

3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\).

10 \includegraphics[max width=\textwidth, alt={}, center]{a49b720b-f8d2-42ff-b147-5d676993aa4c-16_611_689_274_721} The diagram shows the curve \(y = x \cos 2 x\), for \(x \geqslant 0\).

1. Find the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 2 } \pi\).
2. Find the exact area of the shaded region shown in the diagram, bounded by the curve and the \(x\)-axis.

2 Find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln 3 x \mathrm {~d} x\). Give your answer in the form \(a \ln b + c\), where \(a\) and \(c\) are rational and \(b\) is an integer. \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-04_2720_38_105_2010}

7 The integral \(\mathrm { I } _ { \mathrm { n } }\), where n is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 4 } { 3 } } \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { - 1 }\) giving your answer in the form \(\ln a\), where \(a\) is an integer to be determined.
2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm { n } \right)\), or otherwise, show that $$( \mathrm { n } + 1 ) \mathrm { I } _ { \mathrm { n } } = \mathrm { nl } _ { \mathrm { n } - 2 } + \frac { 4 } { 3 } \left( \frac { 5 } { 3 } \right) ^ { \mathrm { n } }$$
3. A curve has equation \(y = x ^ { 2 }\), for \(0 \leqslant x \leqslant \frac { 2 } { 3 }\). The arc length of the curve is denoted by \(s\). Use the substitution \(\mathrm { u } = 2 \mathrm { x }\) to show that \(\mathrm { s } = \frac { 1 } { 2 } \mathrm { l } _ { 1 }\) and find the exact value of \(s\).

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$ The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 3\)

1. Complete the table below with the value of \(y\) corresponding to \(x = 2\). Give your answer to 4 decimal places.

   \(x\)	1	1.5	2	2.5	3
   \(y\)	2	1.3041		0.9089	1.2958

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(S\), giving your answer to 3 decimal places.
3. Use calculus to find the exact area of \(S\). Give your answer in the form \(\frac { a } { b } + \ln c\), where \(a , b\) and \(c\) are integers.
4. Hence calculate the percentage error in using your answer to part (b) to estimate the area of \(S\). Give your answer to one decimal place.
5. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(S\). \section*{Question 12 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 13. (a) Express \(10 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\) Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places. Alana models the height above the ground of a passenger on a Ferris wheel by the equation $$H = 12 - 10 \cos ( 30 t ) ^ { \circ } + 3 \sin ( 30 t ) ^ { \circ }$$ where the height of the passenger above the ground is \(H\) metres at time \(t\) minutes after the wheel starts turning. \includegraphics[max width=\textwidth, alt={}, center]{03548211-79cb-4629-b6ca-aa9dfcc77a33-23_419_567_516_1160}
   (b) Calculate
   1. the maximum value of \(H\) predicted by this model,
   2. the value of \(t\) when this maximum first occurs. Give each answer to 2 decimal places.
      (c) Calculate the value of \(t\) when the passenger is 18 m above the ground for the first time. Give your answer to 2 decimal places.
      (d) Determine the time taken for the Ferris wheel to complete two revolutions. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 13 continued}

8. $$f ( \theta ) = 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta$$

1. Show that \(\mathrm { f } ( \theta ) = a + b \cos 2 \theta\), where \(a\) and \(b\) are integers which should be found.
2. Using your answer to part (a) and integration by parts, find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \theta ^ { 2 } \mathrm { f } ( \theta ) \mathrm { d } \theta$$

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x \cos x , \quad x \in \mathbb { R }$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the \(x\)-axis for \(\frac { 3 \pi } { 2 } \leqslant x \leqslant \frac { 5 \pi } { 2 }\)

1. Complete the table below with the exact value of \(y\) corresponding to \(x = \frac { 7 \pi } { 4 }\) and with the exact value of \(y\) corresponding to \(x = \frac { 9 \pi } { 4 }\)

   \(x\)	\(\frac { 3 \pi } { 2 }\)	\(\frac { 7 \pi } { 4 }\)	\(2 \pi\)	\(\frac { 9 \pi } { 4 }\)	\(\frac { 5 \pi } { 2 }\)
   \(y\)	0		\(2 \pi\)		0

2. Use the trapezium rule, with all five \(y\) values in the completed table, to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.
3. Find $$\int x \cos x d x$$
4. Using your answer from part (c), find the exact area of the region \(R\).

13. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve with equation \(y = 2 - \ln x , x > 0\) The finite region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \mathrm { e }\). The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 - \ln x\)

1. Complete the table giving the value of \(y\) to 4 decimal places.
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 3 decimal places.
3. Use integration by parts to show that \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x = x ( \ln x ) ^ { 2 } - 2 x \ln x + 2 x + c\) The area \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
4. Use calculus to find the exact volume of the solid generated. Write your answer in the form \(\pi \mathrm { e } ( p \mathrm { e } + q )\), where \(p\) and \(q\) are integers to be found.

1. Use integration by parts to find the exact value of \(\int _ { 1 } ^ { \mathrm { e } } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x\)

Write your answer in the form \(a + \frac { b } { \mathrm { e } }\), where \(a\) and \(b\) are integers.

1. (i) Find

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

13. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(y = 12 x ^ { 2 } \ln \left( 2 x ^ { 2 } \right) , x > 0\) The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = 2\) The table below shows corresponding values of \(x\) and \(y\) for \(y = 12 x ^ { 2 } \ln \left( 2 x ^ { 2 } \right)\), with the values of \(y\) given to 3 significant figures.

1. Use the trapezium rule, with all the values of \(y\), to obtain an estimate for the area of \(R\), giving your answer to 2 significant figures.
2. Use the substitution \(u = x ^ { 2 }\) to show that the area of \(R\) is given by $$\int _ { 1 } ^ { 4 } 6 u ^ { \frac { 1 } { 2 } } \ln ( 2 u ) \mathrm { d } u$$
3. Hence, using calculus, find the exact area of \(R\), writing your answer in the form \(a + b \ln 2\), where \(a\) and \(b\) are constants to be found.

8. (i) Find \(\int x \sin x d x\) (ii) (a) Use the substitution \(x = \sec \theta\) to show that
(b) Hence find the exact value of $$\int _ { 1 } ^ { 2 } \sqrt { 1 - \frac { 1 } { x ^ { 2 } } } \mathrm {~d} x = \int _ { 0 } ^ { \frac { \pi } { 3 } } \tan ^ { 2 } \theta \mathrm {~d} \theta$$ Hence find the exact value of $$\int _ { 1 } ^ { 2 } \sqrt { 1 - \frac { 1 } { x ^ { 2 } } } \mathrm {~d} x$$

7. \begin{figure}[h] \end{figure}

1. Find \(\int \mathrm { e } ^ { 2 x } \sin x \mathrm {~d} x\) Figure 2 shows a sketch of part of the curve with equation $$y = \mathrm { e } ^ { 2 x } \sin x \quad x \geqslant 0$$ The finite region \(R\) is bounded by the curve and the \(x\)-axis and is shown shaded in Figure 2.
2. Show that the exact area of \(R\) is \(\frac { \mathrm { e } ^ { 2 \pi } + 1 } { 5 }\) (Solutions relying on calculator technology are not acceptable.)
   Question 7 continue

6. Use integration by parts to show that $$\int \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x = p \mathrm { e } ^ { 2 x } \sin 3 x + q \mathrm { e } ^ { 2 x } \cos 3 x + k$$ where \(p\) and \(q\) are rational numbers to be found and \(k\) is an arbitrary constant.
(6) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 6 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

13
- 1
4 \end{array} \right) + \mu \left( \begin{array} { r } 5
1
- 3 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection \(A\).
2. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to one decimal place. A circle with centre \(A\) and radius 35 cuts the line \(l _ { 1 }\) at the points \(P\) and \(Q\). Given that the \(x\) coordinate of \(P\) is greater than the \(x\) coordinate of \(Q\),
3. find the coordinates of \(P\) and the coordinates of \(Q\). 6. Use integration by parts to show that $$\int \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x = p \mathrm { e } ^ { 2 x } \sin 3 x + q \mathrm { e } ^ { 2 x } \cos 3 x + k$$ where \(p\) and \(q\) are rational numbers to be found and \(k\) is an arbitrary constant.\\ (6)\\ 7. Water is flowing into a large container and is leaking from a hole at the base of the container. At time \(t\) seconds after the water starts to flow, the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the container is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 300 - k V$$ where \(k\) is a constant.
4. Solve the differential equation to show that, according to the model, $$V = \frac { 300 } { k } + A \mathrm { e } ^ { - k t }$$ where \(A\) is a constant.\\ (5) Given that the container is initially empty and that when \(t = 10\), the volume of water is increasing at a rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
5. find the exact value of \(k\).
6. Hence find, according to the model, the time taken for the volume of water in the container to reach 6 litres. Give your answer to the nearest second.\\ 8. Use proof by contradiction to prove that, for all positive real numbers \(x\) and \(y\), $$\frac { 9 x } { y } + \frac { y } { x } \geqslant 6$$ 9. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{594542dd-ee2d-49b6-9fab-77b2d1a44f8c-24_632_734_214_607} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of a closed curve with parametric equations $$x = 5 \cos \theta \quad y = 3 \sin \theta - \sin 2 \theta \quad 0 \leqslant \theta < 2 \pi$$ The region enclosed by the curve is rotated through \(\pi\) radians about the \(x\)-axis to form a solid of revolution.
7. Show that the volume, \(V\), of the solid of revolution is given by $$V = 5 \pi \int _ { \alpha } ^ { \beta } \sin ^ { 3 } \theta ( 3 - 2 \cos \theta ) ^ { 2 } \mathrm {~d} \theta$$ where \(\alpha\) and \(\beta\) are constants to be found.
8. Use the substitution \(u = \cos \theta\) and algebraic integration to show that \(V = k \pi\) where \(k\) is a rational number to be found. \includegraphics[max width=\textwidth, alt={}, center]{594542dd-ee2d-49b6-9fab-77b2d1a44f8c-28_2649_1889_109_178}

7. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 3 shows the design of a doorknob.
The shape of the doorknob is formed by rotating the curve shown in Figure 4 through \(360 ^ { \circ }\) about the \(x\)-axis, where the units are centimetres. The equation of the curve is given by $$\mathrm { f } ( x ) = \frac { 1 } { 4 } ( 4 - x ) \mathrm { e } ^ { x } \quad 0 \leqslant x \leqslant 4$$

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the doorknob is given by $$V = K \int _ { 0 } ^ { 4 } \left( x ^ { 2 } - 8 x + 16 \right) \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ where \(K\) is a constant to be found.
2. Hence, find the exact value of the volume of the doorknob. Give your answer in the form \(p \pi \left( \mathrm { e } ^ { q } + r \right) \mathrm { cm } ^ { 3 }\) where \(p , q\) and \(r\) are simplified rational numbers to be found.

1. (a) Using the substitution \(u = \sqrt { 2 x + 1 }\), show that

$$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } \mathrm { e } ^ { \sqrt { 2 x + 1 } } \mathrm {~d} x$$ may be expressed in the form $$\int _ { a } ^ { b } k u ^ { 2 } \mathrm { e } ^ { u } \mathrm {~d} u$$ where \(a\), \(b\) and \(k\) are constants to be found.
(b) Hence find, by algebraic integration, the exact value of $$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } e ^ { \sqrt { 2 x + 1 } } d x$$ giving your answer in simplest form.