Integration by Parts

3. Find

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

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

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 Find \(\int x \cos 3 x \mathrm {~d} x\).

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

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

2 Find \(\int x ^ { 8 } \ln ( 3 x ) \mathrm { d } x\).

2 Find the exact value of \(\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x\).

4 Show that \(\int _ { 1 } ^ { 4 } x ^ { - \frac { 3 } { 2 } } \ln x \mathrm {~d} x = 2 - \ln 4\).

9 \includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_399_696_255_721} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { 2 - x }\) and its maximum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) is 2 .
2. Find the exact value of \(\int _ { 0 } ^ { 2 } x ^ { 2 } \mathrm { e } ^ { 2 - x } \mathrm {~d} x\).

7 \includegraphics[max width=\textwidth, alt={}, center]{25dffd43-9456-449b-be77-8402109ee603-3_608_672_283_733} The diagram shows the curve \(y = 2 \mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - 2 x }\). The curve cuts the \(y\)-axis at \(A\).

1. Write down the coordinates of \(A\).
2. Find the equation of the tangent to the curve at \(A\), and state the coordinates of the point where this tangent meets the \(x\)-axis.
3. Calculate the area of the region bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = 1\), giving your answer correct to 2 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{f4614578-f5f6-4283-8185-8b5598ad91d5-3_416_679_258_731} The diagram shows part of the curve \(y = \left( 2 x - x ^ { 2 } \right) \mathrm { e } ^ { \frac { 1 } { 2 } x }\) and its maximum point \(M\).

1. Find the exact \(x\)-coordinate of \(M\).
2. Find the exact value of the area of the shaded region bounded by the curve and the positive \(x\)-axis.

2. Show that $$\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x = 2 \ln 2 - \frac { 3 } { 4 }$$

2. Show that $$\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x = 2 \ln 2 - \frac { 3 } { 4 }$$

7 Show that \(\int _ { 0 } ^ { \pi } \left( x ^ { 2 } + 5 x + 7 \right) \sin x \mathrm {~d} x = \pi ^ { 2 } + 5 \pi + 10\).

4

1. Use integration by parts to find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
2. Hence find \(\int x \tan ^ { 2 } x \mathrm {~d} x\).

2. (a) Use integration by parts to find \(\int x \mathrm { e } ^ { x } \mathrm {~d} x\).
(b) Hence find \(\int x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).

6. (a) Use the substitution \(u = \sqrt { t }\) to show that $$\int _ { 1 } ^ { x } \frac { \ln t } { \sqrt { t } } \mathrm {~d} t = 4 - 4 \sqrt { x } + 2 \sqrt { x } \ln x \quad x \geqslant 1$$ (b) The function g is such that $$\int _ { 1 } ^ { x } \mathrm {~g} ( t ) \mathrm { d } t = x - \sqrt { x } \ln x - 1 \quad x \geqslant 1$$

1. Use differentiation to find the function g .
2. Evaluate \(\int _ { 4 } ^ { 16 } \mathrm {~g} ( t ) \mathrm { d } t\) and simplify your answer.
   (c) Find the value of \(x\) (where \(x > 1\) ) that gives the maximum value of $$\int _ { x } ^ { x + 1 } \frac { \ln t } { 2 ^ { t } } \mathrm {~d} t$$

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 \(I _ { 2 }\)

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

1. Show that $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { x } \cos x \mathrm {~d} x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \frac { 1 } { 2 } \pi } + \mathrm { e } ^ { - \frac { 1 } { 2 } \pi } \right)$$
2. It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos ^ { n } x \mathrm {~d} x$$ Show that, for \(n \geqslant 2\), $$4 I _ { n } = n ( n - 1 ) \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin ^ { 2 } x \cos ^ { n - 2 } x \mathrm {~d} x - n I _ { n }$$ and deduce the reduction formula $$\left( n ^ { 2 } + 4 \right) I _ { n } = n ( n - 1 ) I _ { n - 2 }$$
3. Using the result in part (i) and the reduction formula in part (ii), find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis and the arc of the curve \(y = \mathrm { e } ^ { x } \cos x\) from \(x = - \frac { 1 } { 2 } \pi\) to \(x = \frac { 1 } { 2 } \pi\). Give your answer correct to 3 significant figures.

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.

2

1. Find \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), where \(a > 0\).
2. Write down the value of \(\lim _ { a \rightarrow \infty } a ^ { k } \mathrm { e } ^ { - 2 a }\), where \(k\) is a positive constant.
3. Hence find \(\int _ { 0 } ^ { \infty } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).

7

1. Write down the value of $$\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x }$$
2. Use the substitution \(u = x \mathrm { e } ^ { - x } + 1\) to find \(\int \frac { \mathrm { e } ^ { - x } ( 1 - x ) } { x \mathrm { e } ^ { - x } + 1 } \mathrm {~d} x\).
3. Hence evaluate \(\int _ { 1 } ^ { \infty } \frac { 1 - x } { x + \mathrm { e } ^ { x } } \mathrm {~d} x\), showing the limiting process used.

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

2 Show that \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x = \pi ^ { 2 } - 4\).

1. Use integration by parts to find

$$\int x ^ { 2 } \sin x d x$$

8 \includegraphics[max width=\textwidth, alt={}, center]{dde12c57-5129-43ae-b385-9a8f21f51e49-3_566_787_255_680} The diagram shows the curve \(y = x \sin x\), for \(0 \leqslant x \leqslant \pi\). The point \(Q \left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi \right)\) lies on the curve.

1. Show that the normal to the curve at \(Q\) passes through the point \(( \pi , 0 )\).
2. Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin x - x \cos x )\).
3. Hence evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).

10 \includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-18_337_529_260_808} The diagram shows the curve \(y = ( \ln x ) ^ { 2 }\). The \(x\)-coordinate of the point \(P\) is equal to e, and the normal to the curve at \(P\) meets the \(x\)-axis at \(Q\).

1. Find the \(x\)-coordinate of \(Q\).
2. Show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\), where \(c\) is a constant.
3. Using integration by parts, or otherwise, find the exact value of the area of the shaded region between the curve, the \(x\)-axis and the normal \(P Q\).

9. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x , \quad x > 0\) 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 \(A \mathrm { e } ^ { 2 } + B\) where \(A\) and \(B\) are rational numbers to be found.
[0pt] [BLANK PAGE]

4

1. Differentiate \(\mathrm { e } ^ { x } ( \sin 2 x - 2 \cos 2 x )\), simplifying your answer.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \mathrm { e } ^ { x } \sin 2 x \mathrm {~d} x\).

5 It is given that \(\int _ { 1 } ^ { a } x \ln x \mathrm {~d} x = 22\), where \(a\) is a constant greater than 1 .

1. Show that \(a = \sqrt { } \left( \frac { 87 } { 2 \ln a - 1 } \right)\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 6 and give the result of each iteration to 4 decimal places.

9 \includegraphics[max width=\textwidth, alt={}, center]{bbc19395-6f88-4a7c-b5d4-59ced9ccdcf2-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) 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 \(x = 2\).

2．Given that \(S = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { 2 x } \sin x \mathrm {~d} x\) and \(C = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { 2 x } \cos x \mathrm {~d} x\) ，
（a）show that \(S = 1 + 2 C\) ，
（b）find the exact value of \(S\) ．

6

1. By using integration by parts twice, find $$\int x ^ { 2 } \sin 2 x d x$$
2. A curve has equation \(y = x \sqrt { \sin 2 x }\), for \(0 \leqslant x \leqslant \frac { \pi } { 2 }\). The region bounded by the curve and the \(x\)-axis is rotated through \(2 \pi\) radians about the \(x\)-axis to generate a solid. Find the exact value of the volume of the solid generated.
   [0pt] [3 marks]

6. The region bounded by the curve $$y = ( 2 x - 8 ) \ln x$$ and the \(x\)-axis is shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-12_871_913_422_575} Show that the exact area is given by $$32 \ln 2 - \frac { 33 } { 2 }$$ Fully justify your answer.

4 Using integration by parts, find the exact value of \(\int _ { 0 } ^ { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).

7 The integral \(I\) is defined by \(I = \int _ { 0 } ^ { 2 } 4 t ^ { 3 } \ln \left( t ^ { 2 } + 1 \right) \mathrm { d } t\).

1. Use the substitution \(x = t ^ { 2 } + 1\) to show that \(I = \int _ { 1 } ^ { 5 } ( 2 x - 2 ) \ln x \mathrm {~d} x\).
2. Hence find the exact value of \(I\).

8

1. Show that \(\int \cos ^ { 2 } 6 x \mathrm {~d} x = \frac { 1 } { 2 } x + \frac { 1 } { 24 } \sin 12 x + c\).
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } x \cos ^ { 2 } 6 x \mathrm {~d} x\).

2 Use integration by parts to find \(\int \ln ( x + 2 ) \mathrm { d } x\).