1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)

8 In this question you must show detailed reasoning. Given that \(\int _ { 4 } ^ { a } \left( \frac { 4 } { \sqrt { x } } + 3 \right) \mathrm { d } x = 7\), find the value of \(a\).

8

1. Given that \(y = \sec x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
2. In this question you must show detailed reasoning. Find the exact value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 6 } \pi } ( \sec 2 x + \tan 2 x ) ^ { 2 } \mathrm {~d} x\).

1

1. Differentiate the following with respect to \(x\).
   1. \(\frac { 1 } { ( 3 x - 4 ) ^ { 2 } }\)
   2. \(\frac { \ln ( x + 2 ) } { x }\)
   3. Find \(\int \mathrm { e } ^ { ( 2 x + 3 ) } \mathrm { d } x\).

8

1. Write down \(\int \sec ^ { 2 } x \mathrm {~d} x\).
2. Given that \(y = \frac { \cos x } { \sin x }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
3. Prove the identity \(( \tan x + \cot x ) ^ { 2 } = \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x\).
4. Hence find \(\int _ { 0.5 } ^ { 1 } ( \tan x + \cot x ) ^ { 2 } \mathrm {~d} x\), giving your answer to two significant figures.

6

1. Express \(\cos 2 x\) in the form \(a \cos ^ { 2 } x + b\), where \(a\) and \(b\) are constants.
2. Hence show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } x \mathrm {~d} x = \frac { \pi } { a }\), where \(a\) is an integer.

3

1. Express \(\cos 2 x\) in terms of \(\sin x\).
2. 1. Hence show that \(3 \sin x - \cos 2 x = 2 \sin ^ { 2 } x + 3 \sin x - 1\) for all values of \(x\).
   2. Solve the equation \(3 \sin x - \cos 2 x = 1\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
3. Use your answer from part (a) to find \(\int \sin ^ { 2 } x \mathrm {~d} x\).

3

1. 1. Express \(\frac { 2 x + 7 } { x + 2 }\) in the form \(A + \frac { B } { x + 2 }\), where \(A\) and \(B\) are integers. (2 marks)
   2. Hence find \(\int \frac { 2 x + 7 } { x + 2 } \mathrm {~d} x\).
2. 1. Express \(\frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } }\) in the form \(\frac { P } { 1 + 3 x } + \frac { Q } { 5 - x } + \frac { R } { ( 5 - x ) ^ { 2 } }\), where \(P , Q\) and \(R\) are constants.
   2. Hence find \(\int \frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } } \mathrm {~d} x\).

4 The expression \(\frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) }\) can be written in the form \(2 + \frac { A } { x + 1 } + \frac { B } { 5 x - 1 }\), where \(A\) and \(B\) are constants.

1. Find the values of \(A\) and \(B\).
2. Hence find \(\int \frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) } \mathrm { d } x\).

2

1. Express \(\frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) }\) in the form \(\frac { A } { x + 3 } + \frac { B } { 2 x - 1 }\).
2. Hence find \(\int \frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) } \mathrm { d } x\).

6

1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
2. Using the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) :
   1. express \(\cos 2 x\) in terms of \(\sin x\) and \(\cos x\);
   2. show, by writing \(3 x\) as \(( 2 x + x )\), that $$\cos 3 x = 4 \cos ^ { 3 } x - 3 \cos x$$
3. Show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 3 } x \mathrm {~d} x = \frac { 2 } { 3 }\).

3

1. By writing \(\sin 3 x\) as \(\sin ( x + 2 x )\), show that \(\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x\) for all values of \(x\).
2. Hence, or otherwise, find \(\int \sin ^ { 3 } x \mathrm {~d} x\).

1 A particle moves along a straight line. At time \(t\), it has velocity \(v\), where $$v = 4 t ^ { 3 } - 8 \sin 2 t + 5$$ When \(t = 0\), the particle is at the origin.
Find an expression for the displacement of the particle from the origin at time \(t\).

8

1. Hooke's law states that the tension in a stretched string of natural length \(l\) and modulus of elasticity \(\lambda\) is \(\frac { \lambda x } { l }\) when its extension is \(x\). Using this formula, prove that the work done in stretching a string from an unstretched position to a position in which its extension is \(e\) is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
   (3 marks)
2. A particle, of mass 5 kg , is attached to one end of a light elastic string of natural length 0.6 metres and modulus of elasticity 150 N . The other end of the string is fixed to a point \(O\).
   1. Find the extension of the elastic string when the particle hangs in equilibrium directly below \(O\).
   2. The particle is pulled down and held at the point \(P\), which is 0.9 metres vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 11.25 J .
   3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres above \(\boldsymbol { P }\). Show that, while the string is taut, $$v ^ { 2 } = 10.4 x - 50 x ^ { 2 }$$
   4. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.

3

1. Amaya and Ben integrated \(( 1 + x ) ^ { 2 }\), with respect to \(x\), using different methods, as follows. Amaya: \(\quad \int ( 1 + x ) ^ { 2 } \mathrm {~d} x = \frac { ( 1 + x ) ^ { 3 } } { 3 } + c \quad = \frac { 1 } { 3 } + x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + c\) Ben: \(\quad \int ( 1 + x ) ^ { 2 } \mathrm {~d} x = \int \left( 1 + 2 x + x ^ { 2 } \right) \mathrm { d } x = x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + c\) Charlie said that, because these answers are different, at least one of them must be wrong. Explain whether you agree with Charlie's statement.
2. You are given that \(a\) is a constant greater than 1 .
   1. Find \(\int _ { 1 } ^ { a } \frac { 1 } { ( 1 + x ) ^ { 2 } } \mathrm {~d} x\), giving your answer as a single fraction in terms of the constant \(a\).
   2. You are given that the area enclosed by the curve \(y = \frac { 1 } { ( 1 + x ) ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = a\) is equal to \(\frac { 1 } { 3 }\). Determine the value of \(a\).
3. In this question you must show detailed reasoning. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } \frac { \cos 2 x } { \sin 2 x + 2 } \mathrm {~d} x\), giving your answer in its simplest form.

6 Four students, Tom, Josh, Floella and Georgia are attempting to complete the indefinite integral $$\int \frac { 1 } { x } \mathrm {~d} x \quad \text { for } x > 0$$ Each of the students' solutions is shown below: $$\begin{array} { l l } \text { Tom } & \int \frac { 1 } { x } \mathrm {~d} x = \ln x \\ \text { Josh } & \int \frac { 1 } { x } \mathrm {~d} x = k \ln x \\ \text { Floella } & \int \frac { 1 } { x } \mathrm {~d} x = \ln A x \\ \text { Georgia } & \int \frac { 1 } { x } \mathrm {~d} x = \ln x + c \end{array}$$ 6

1. 1. Explain what is wrong with Tom's answer. 6
      1. (ii) Explain what is wrong with Josh's answer.
         6
   2. Explain why Floella and Georgia's answers are equivalent.

14

1. 1. Given that $$y = 2 ^ { x }$$ write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) 14
      1. (ii) Hence find $$\int 2 ^ { x } \mathrm {~d} x$$ 14
   2. The area, \(A\), bounded by the curve with equation \(y = 2 ^ { x }\), the \(x\)-axis, the \(y\)-axis and the line \(x = - 4\) is approximated using eight rectangles of equal width as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-23_1319_978_450_532} 14
   3. 1. Show that the exact area of the largest rectangle is \(\frac { \sqrt { 2 } } { 4 }\) 14
   4. (ii) The areas of these rectangles form a geometric sequence with common ratio \(\frac { \sqrt { 2 } } { 2 }\) Find the exact value of the total area of the eight rectangles.
      Give your answer in the form \(k ( 1 + \sqrt { 2 } )\) where \(k\) is a rational number.
      [0pt] [3 marks]
      14
   5. (iii) More accurate approximations for \(A\) can be found by increasing the number, \(n\), of rectangles used. Find the exact value of the limit of the approximations for \(A\) as \(n \rightarrow \infty\)

5 A train of mass 10000 kg is travelling at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with a buffer. The buffer brings the train to rest. As the buffer brings the train to rest it compresses by 0.2 metres.
When the buffer is compressed by a distance of \(x\) metres it exerts a force of magnitude \(F\) newtons, where $$F = A x + 9000 x ^ { 2 }$$ where \(A\) is a constant. 5

1. Find, in terms of \(A\), the work done in compressing the buffer by 0.2 metres.
   5
2. Find the value of \(A\)

8 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + k x } { x + 1 }\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes.

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve with parametric equations $$x = 3 t ^ { 2 } \quad y = \sin t \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve and the \(x\)-axis.

1. Show that the area of \(R\) is $$k \int _ { 0 } ^ { \frac { \pi } { 2 } } t \sin ^ { 2 } t \cos t \mathrm {~d} t$$ where \(k\) is a constant to be found.
2. Hence, using algebraic integration, find the exact area of \(R\), giving your answer in the form $$p \pi + q$$ where \(p\) and \(q\) are constants.

5

1. Find \(\int \left( \frac { 1 } { x - 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x\) giving your answer in its simplest form.
2. Use integration by parts to find \(\int x ^ { 2 } \ln x \mathrm {~d} x\).

4 Find

1. \(\quad \int ( 2 x + 3 ) ^ { 4 } \mathrm {~d} x\)
2. \(\quad \int \left( 1 + \tan ^ { 2 } 2 x \right) \mathrm { d } x\)