1.08i Integration by parts

5

1. Find \(\int x ^ { 3 } \ln x \mathrm {~d} x\).
2. Explain why \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x\) is an improper integral.
3. Evaluate \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x\), showing the limiting process used.

3

1. Explain why \(\int _ { 1 } ^ { \infty } 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\) is an improper integral.
2. Find \(\quad \int 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\).
3. Hence evaluate \(\int _ { 1 } ^ { \infty } 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\), showing the limiting process used.

3

1. Find \(\int x ^ { 2 } \ln x \mathrm {~d} x\).
2. Explain why \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x\) is an improper integral.
3. Evaluate \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x\), showing the limiting process used.

5

1. Find \(\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\).
2. Hence evaluate \(\int _ { 0 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\), showing the limiting process used.

5

1. Find \(\int x \cos 8 x \mathrm {~d} x\).
2. Find \(\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x } \sin 2 x \right]\).
3. Explain why \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 2 \cot 2 x - \frac { 1 } { x } + x \cos 8 x \right) \mathrm { d } x\) is an improper integral.
4. Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 2 \cot 2 x - \frac { 1 } { x } + x \cos 8 x \right) \mathrm { d } x\), showing the limiting process used. Give your answer as a single term.
   [0pt] [4 marks]

11
The diagram shows part of the curve \(y = \ln ( x - 4 )\).

1. Use integration by parts to show that \(\int \ln ( x - 4 ) \mathrm { d } x = ( x - 4 ) \ln | x - 4 | - x + c\).
2. State the equation of the vertical asymptote to the curve \(y = \ln ( x - 4 )\).
3. Find the total area enclosed by the curve \(y = \ln ( x - 4 )\), the \(x\)-axis and the lines \(x = 4.5\) and \(x = 7\). Give your answer in the form \(a \ln 3 + b \ln 2 + c\) where \(a , b\) and \(c\) are constants to be found.

10 \includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-7_352_545_258_239} The diagram shows the curve \(y = \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right)\), for \(1 \leqslant x \leqslant 2\).

1. Use rectangles of width 0.25 to find upper and lower bounds for \(\int _ { 1 } ^ { 2 } \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x\). Give your answers correct to 3 significant figures.
2. 1. Use the substitution \(t = \sqrt { x - 1 }\) to show that \(\int \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x = \int 2 t \sin \left( \frac { 1 } { 2 } t \right) \mathrm { d } t\).
   2. Hence show that \(\int _ { 1 } ^ { 2 } \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x = 8 \sin \frac { 1 } { 2 } - 4 \cos \frac { 1 } { 2 }\).

12. \begin{figure}[h] \end{figure} The figure 6 shows a sketch of the curve with equation $$y = x ^ { 2 } \ln 2 x$$ The finite region \(R\), shown shaded in figure 5, is bounded by the line with equation \(x = \frac { e ^ { 2 } } { 2 }\), the curve \(C\), the line with equation \(x = e ^ { 2 }\) and the \(x\)-axis.
Show that the exact value of the area of region \(R\) is \(\frac { e ^ { 6 } } { 72 } ( 35 + 24 \ln 2 )\).

1. a. Find \(\int \ln x \mathrm {~d} x\)

\begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = \ln x , \quad x > 0$$ The point P lies on \(C\) and has coordinate \(( e , 1 )\).
The line 1 is a normal to \(C\) at \(P\). The line \(l\) cuts the \(x\)-axis at the point \(Q\).
b. Find the exact value of the \(x\)-coordinate of \(Q\). The finite region \(\mathbf { R }\), shown shaded in figure 3, is bounded by the curve, the line \(l\) and the \(x\)-axis.
c. Find the exact area of \(\mathbf { R }\).

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

Show that $$\int _ { 1 } ^ { \mathrm { e } ^ { 2 } } x ^ { 3 } \ln x \mathrm {~d} x = a \mathrm { e } ^ { 8 } + b$$ where \(a\) and \(b\) are rational constants to be found.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Given that \(1 + \cos 2 \theta + \sin 2 \theta \neq 0\) prove that $$\frac { 1 - \cos 2 \theta + \sin 2 \theta } { 1 + \cos 2 \theta + \sin 2 \theta } \equiv \tan \theta$$
2. Hence solve, for \(0 < x < 180 ^ { \circ }\) $$\frac { 1 - \cos 4 x + \sin 4 x } { 1 + \cos 4 x + \sin 4 x } = 3 \sin 2 x$$ giving your answers to one decimal place where appropriate.

7. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = 2 \mathrm { e } ^ { 2 x } - x \mathrm { e } ^ { 2 x } , \quad x \in \mathbb { R }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(p \mathrm { e } ^ { 4 } + q\), where \(p\) and \(q\) are rational constants to be found.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

13. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x , x > 0\) The line \(l\) is the normal to \(C\) at the point \(P ( \mathrm { e } , \mathrm { e } )\) The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis.
Show that the exact area of \(R\) is \(A e ^ { 2 } + B\) where \(A\) and \(B\) are rational numbers to be found.
(10)

11. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve \(C\) with equation $$y = 8 x ^ { 2 } \mathrm { e } ^ { - 3 x } \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 5, is bounded by

• the curve \(C\)
• the line with equation \(x = 1\)
• the \(x\)-axis

Find the exact area of \(R\), giving your answer in the form $$A + B \mathrm { e } ^ { - 3 }$$ where \(A\) and \(B\) are rational numbers to be found.

1. A medical researcher is studying the number of hours, \(T\), a patient stays in hospital following a particular operation.

The histogram on the page opposite summarises the results for a random sample of 90 patients.

1. Use the histogram to estimate \(\mathrm { P } ( 10 < T < 30 )\) For these 90 patients the time spent in hospital following the operation had
   • a mean of 14.9 hours
   • a standard deviation of 9.3 hours
   Tomas suggests that \(T\) can be modelled by \(\mathrm { N } \left( 14.9,9.3 ^ { 2 } \right)\)
2. With reference to the histogram, state, giving a reason, whether or not Tomas' model could be suitable. Xiang suggests that the frequency polygon based on this histogram could be modelled by a curve with equation $$y = k x \mathrm { e } ^ { - x } \quad 0 \leqslant x \leqslant 4$$ where
   • \(x\) is measured in tens of hours
   • \(k\) is a constant
   • Use algebraic integration to show that
   $$\int _ { 0 } ^ { n } x \mathrm { e } ^ { - x } \mathrm {~d} x = 1 - ( n + 1 ) \mathrm { e } ^ { - n }$$
3. Show that, for Xiang's model, \(k = 99\) to the nearest integer.
4. Estimate \(\mathrm { P } ( 10 < T < 30 )\) using
   1. Tomas' model of \(T \sim \mathrm {~N} \left( 14.9,9.3 ^ { 2 } \right)\)
   2. Xiang's curve with equation \(y = 99 x \mathrm { e } ^ { - x }\) and the answer to part (c) The researcher decides to use Xiang's curve to model \(\mathrm { P } ( a < T < b )\)
5. State one limitation of Xiang's model. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Question 6 continued} \includegraphics[alt={},max width=\textwidth]{a067577e-e2a6-440b-9d22-d558fade15f0-17_1164_1778_294_146}
   \end{figure} Time in hours

10 Fig. 10 shows the graph of \(y = ( k - x ) \ln x\) where \(k\) is a constant ( \(k > 1\) ). \begin{figure}[h] \end{figure} Find, in terms of \(k\), the area of the finite region between the curve and the \(x\)-axis.

8 Find \(\int x ^ { 2 } \mathrm { e } ^ { 2 x } \mathrm {~d} x\).

6 Find \(\int \frac { 32 } { x ^ { 5 } } \ln x \mathrm {~d} x\). Answer all the questions
Section B (78 marks)

10 Determine the exact value of \(\int _ { 0 } ^ { \frac { \pi } { 4 } } 4 x \cos 2 x d x\).

7 Determine \(\int x \cos 2 x \mathrm {~d} x\).

7 Let \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 2 } \frac { \mathrm { x } ^ { \mathrm { n } } } { \sqrt { \mathrm { x } ^ { 3 } + 1 } } \mathrm { dx }\) for integers \(n > 0\).

1. By considering the derivative of \(\sqrt { x ^ { 3 } + 1 }\) with respect to \(x\), determine the exact value of \(I _ { 2 }\).
2. Given that \(n > 3\), show that \(\left. ( 2 n - 1 ) \right| _ { n } = 3 \times 2 ^ { n - 1 } - \left. 2 ( n - 2 ) \right| _ { n - 3 }\).
3. Hence determine the exact value of \(\int _ { 0 } ^ { 2 } x ^ { 5 } \sqrt { x ^ { 3 } + 1 } \mathrm {~d} x\).

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.

8

1. Given that \(\mathrm { e } ^ { - 2 x } = 3\), find the exact value of \(x\).
2. Use integration by parts to find \(\int x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
3. A curve has equation \(y = \mathrm { e } ^ { - 2 x } + 6 x\).
   1. Find the exact values of the coordinates of the stationary point of the curve.
   2. Determine the nature of the stationary point.
   3. The region \(R\) is bounded by the curve \(y = \mathrm { e } ^ { - 2 x } + 6 x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid formed when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer to three significant figures.

5

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

7

1. A curve has equation \(y = x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } }\).
   Show that the curve has exactly two stationary points and find the exact values of their coordinates.
   (7 marks)
2. 1. Use integration by parts twice to find the exact value of \(\int _ { 0 } ^ { 4 } x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } } \mathrm {~d} x\).
   2. The region bounded by the curve \(y = 3 x \mathrm { e } ^ { - \frac { x } { 8 } }\), the \(x\)-axis from 0 to 4 and the line \(x = 4\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid. Use your answer to part (b)(i) to find the exact value of the volume of the solid generated.