Questions — Edexcel P4 (127 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Edexcel P4 2020 October Q5
7 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{79ac81c3-cd05-4f28-8840-3c8a6960e7b7-14_600_1022_255_461} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
  1. Find \(\int \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x\) Figure 3 shows a sketch of part of the curve with equation $$y = \frac { 3 + 2 x + \ln x } { x ^ { 2 } } \quad x > 0.5$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 4\)
  2. Use the answer to part (a) to find the exact area of \(R\), writing your answer in simplest form.
Edexcel P4 2020 October Q6
7 marks Standard +0.8
6. A curve \(C\) has equation $$y = x ^ { \sin x } \quad x > 0 \quad y > 0$$
  1. Find, by firstly taking natural logarithms, an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Hence show that the \(x\) coordinates of the stationary points of \(C\) are solutions of the equation $$\tan x + x \ln x = 0$$
Edexcel P4 2020 October Q7
12 marks Standard +0.3
7. (i) Using a suitable substitution, find, using calculus, the value of $$\int _ { 1 } ^ { 5 } \frac { 3 x } { \sqrt { 2 x - 1 } } \mathrm {~d} x$$ (Solutions relying entirely on calculator technology are not acceptable.)
(ii) Find $$\int \frac { 6 x ^ { 2 } - 16 } { ( x + 1 ) ( 2 x - 3 ) } d x$$
Edexcel P4 2020 October Q8
10 marks Standard +0.3
8. Relative to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 4 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ - 2 \\ - 1 \end{array} \right) \quad \text { where } \lambda \text { is a scalar parameter } \\ & l _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { r } 2 \\ 0 \\ - 9 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ - 1 \\ - 3 \end{array} \right) \quad \text { where } \mu \text { is a scalar parameter } \end{aligned}$$ Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(X\),
  1. find the position vector of \(X\). The point \(P ( 10 , - 7,0 )\) lies on \(l _ { 1 }\) The point \(Q\) lies on \(l _ { 2 }\) Given that \(\overrightarrow { P Q }\) is perpendicular to \(l _ { 2 }\)
  2. calculate the coordinates of \(Q\).
Edexcel P4 2020 October Q9
9 marks Moderate -0.3
9. Bacteria are growing on the surface of a dish in a laboratory. The area of the dish, \(A \mathrm {~cm} ^ { 2 }\), covered by the bacteria, \(t\) days after the bacteria start to grow, is modelled by the differential equation $$\frac { \mathrm { d } A } { \mathrm {~d} t } = \frac { A ^ { \frac { 3 } { 2 } } } { 5 t ^ { 2 } } \quad t > 0$$ Given that \(A = 2.25\) when \(t = 3\)
  1. show that $$A = \left( \frac { p t } { q t + r } \right) ^ { 2 }$$ where \(p , q\) and \(r\) are integers to be found. According to the model, there is a limit to the area that will be covered by the bacteria.
  2. Find the value of this limit. \includegraphics[max width=\textwidth, alt={}, center]{79ac81c3-cd05-4f28-8840-3c8a6960e7b7-31_2255_50_314_34}
    VIIV SIHI NI JIIIM ION OCVIAV SIHI NI I II M I I O N OCVAYV SIHI NI JIIIM ION OO
Edexcel P4 2021 October Q1
7 marks Standard +0.3
  1. The curve \(C\) has equation
$$2 x - 4 y ^ { 2 } + 3 x ^ { 2 } y = 4 x ^ { 2 } + 8$$ The point \(P ( 3,2 )\) lies on \(C\).
Find the equation of the normal to \(C\) at the point \(P\), writing your answer in the form \(a x + b y + c = 0\) where \(a\), \(b\) and \(c\) are integers to be found.
Edexcel P4 2021 October Q2
6 marks Moderate -0.3
2. Find the particular solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y ^ { 2 } } { \sqrt { 4 x + 5 } } \quad x > - \frac { 5 } { 4 }$$ for which \(y = \frac { 1 } { 3 }\) at \(x = - \frac { 1 } { 4 }\) giving your answer in the form \(y = \mathrm { f } ( x )\) (6)
Edexcel P4 2021 October Q3
8 marks Standard +0.3
3. $$\mathrm { g } ( x ) = \frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 6 } { x ( x + 3 ) } \equiv A x + B + \frac { C } { x } + \frac { D } { x + 3 }$$
  1. Find the values of the constants \(A , B , C\) and \(D\). A curve has equation $$y = g ( x ) \quad x > 0$$ Using the answer to part (a),
  2. find \(\mathrm { g } ^ { \prime } ( x )\).
  3. Hence, explain why \(\mathrm { g } ^ { \prime } ( x ) > 3\) for all values of \(x\) in the domain of g .
Edexcel P4 2021 October Q4
6 marks Standard +0.3
4. $$\mathrm { f } ( x ) = \sqrt { 1 - 4 x ^ { 2 } } \quad | x | < \frac { 1 } { 2 }$$
  1. Find, in ascending powers of \(x\), the first four non-zero terms of the binomial expansion of \(\mathrm { f } ( x )\). Give each coefficient in simplest form.
  2. By substituting \(x = \frac { 1 } { 4 }\) into the binomial expansion of \(\mathrm { f } ( x )\), obtain an approximation for \(\sqrt { 3 }\) Give your answer to 4 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{08756c4b-6619-42da-ac8a-2bf065c01de8-13_42_63_2606_1852}
Edexcel P4 2021 October Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-14_787_638_251_653} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = 5 + 2 \tan t \quad y = 8 \sec ^ { 2 } t \quad - \frac { \pi } { 3 } \leqslant t \leqslant \frac { \pi } { 4 }$$
  1. Use parametric differentiation to find the gradient of \(C\) at \(x = 3\) The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where f is a quadratic function.
  2. Find \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are constants to be found.
  3. Find the range of f.
Edexcel P4 2021 October Q6
7 marks Challenging +1.2
6. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-18_650_938_413_504} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 16 \sin 2 x } { ( 3 + 4 \sin x ) ^ { 2 } } \quad 0 \leqslant x \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { \pi } { 6 }\) Using the substitution \(u = 3 + 4 \sin x\), show that the area of \(R\) can be written in the form \(a + \ln b\), where \(a\) and \(b\) are rational constants to be found.
Edexcel P4 2021 October Q7
9 marks Standard +0.8
7. With respect to a fixed origin \(O\),
  • the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 4 \\ 2 \\ - 3 \end{array} \right) + \lambda \left( \begin{array} { r } - 4 \\ - 3 \\ 5 \end{array} \right)\) where \(\lambda\) is a scalar constant
  • the point \(A\) has position vector \(9 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\)
Given that \(X\) is the point on \(l\) nearest to \(A\),
  1. find
    1. the coordinates of \(X\)
    2. the shortest distance from \(A\) to \(l\). Give your answer in the form \(\sqrt { d }\), where \(d\) is an integer. The point \(B\) is the image of \(A\) after reflection in \(l\).
  2. Find the position vector of \(B\). Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-26_668_661_408_644} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure}
Edexcel P4 2021 October Q9
10 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-30_528_1031_242_452} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a cylindrical tank that contains some water. The tank has an internal diameter of 8 m and an internal height of 4.2 m .
Water is flowing into the tank at a constant rate of \(( 0.6 \pi ) \mathrm { m } ^ { 3 }\) per minute. There is a tap at point \(T\) at the bottom of the tank. At time \(t\) minutes after the tap has been opened,
  • the depth of the water is \(h\) metres
  • the water is leaving the tank at a rate of \(( 0.15 \pi h ) \mathrm { m } ^ { 3 }\) per minute
    1. Show that
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 12 - 3 h } { 320 }$$ Given that the depth of the water in the tank is 0.5 m when the tap is opened,
  • find the time taken for the depth of water in the tank to reach 3.5 m .
    •
    Edexcel P4 2021 October Q10
    6 marks Standard +0.3
    10. (a) A student's attempt to answer the question
    "Prove by contradiction that if \(n ^ { 3 }\) is even, then \(n\) is even" is shown below. Line 5 of the proof is missing. Assume that there exists a number \(n\) such that \(n ^ { 3 }\) is even, but \(n\) is odd. If \(n\) is odd then \(n = 2 p + 1\) where \(p \in \mathbb { Z }\) So \(n ^ { 3 } = ( 2 p + 1 ) ^ { 3 }\) $$\begin{aligned} & = 8 p ^ { 3 } + 12 p ^ { 2 } + 6 p + 1 \\ & = \end{aligned}$$ This contradicts our initial assumption, so if \(n ^ { 3 }\) is even, then \(n\) is even. Complete this proof by filling in line 5.
    (b) Hence, prove by contradiction that \(\sqrt [ 3 ] { 2 }\) is irrational.
    Edexcel P4 2023 October Q1
    5 marks Moderate -0.3
    1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of
    $$\frac { 8 } { ( 2 - 5 x ) ^ { 2 } }$$ writing each term in simplest form.
    (b) Find the range of values of \(x\) for which this expansion is valid.
    Edexcel P4 2023 October Q2
    7 marks Standard +0.3
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7f5fc83d-ab7c-4edb-a2c6-7a58f1357d5a-04_271_223_246_922} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a cube which is increasing in size.
    At time \(t\) seconds,
    • the length of each edge of the cube is \(x \mathrm {~cm}\)
    • the surface area of the cube is \(S \mathrm {~cm} ^ { 2 }\)
    • the volume of the cube is \(V \mathrm {~cm} ^ { 3 }\)
    Given that the surface area of the cube is increasing at a constant rate of \(4 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\)
    1. show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { k } { x }\) where \(k\) is a constant to be found,
    2. show that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = V ^ { p }\) where \(p\) is a constant to be found.
    Edexcel P4 2023 October Q3
    12 marks Standard +0.3
    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.
    Edexcel P4 2023 October Q4
    5 marks Moderate -0.3
    1. (a) Prove by contradiction that for all positive numbers \(k\)
    $$k + \frac { 9 } { k } \geqslant 6$$ (b) Show that the result in part (a) is not true for all real numbers.
    Edexcel P4 2023 October Q5
    10 marks Standard +0.8
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7f5fc83d-ab7c-4edb-a2c6-7a58f1357d5a-12_678_987_248_539} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$y ^ { 3 } - x ^ { 2 } + 4 x ^ { 2 } y = k$$ where \(k\) is a positive constant greater than 1
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) lies on \(C\).
      Given that the normal to \(C\) at \(P\) has equation \(y = x\), as shown in Figure 2,
    2. find the value of \(k\).
    Edexcel P4 2023 October Q6
    10 marks Standard +0.8
    1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 7 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 2 \end{array} \right)\) where \(\lambda\) is a scalar parameter.
    The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { r } 4 \\ - 1 \\ 8 \end{array} \right)\) where \(\mu\) is a scalar parameter.
    Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\)
    1. state the coordinates of \(P\) Given that the angle between lines \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\)
    2. find the value of \(\cos \theta\), giving the answer as a fully simplified fraction. The point \(Q\) lies on \(l _ { 1 }\) where \(\lambda = 6\) Given that point \(R\) lies on \(l _ { 2 }\) such that triangle \(Q P R\) is an isosceles triangle with \(P Q = P R\)
    3. find the exact area of triangle \(Q P R\)
    4. find the coordinates of the possible positions of point \(R\)
    Edexcel P4 2023 October Q7
    12 marks Standard +0.3
    1. The number of goats on an island is being monitored.
    When monitoring began there were 3000 goats on the island.
    In a simple model, the number of goats, \(x\), in thousands, is modelled by the equation $$x = \frac { k ( 9 t + 5 ) } { 4 t + 3 }$$ where \(k\) is a constant and \(t\) is the number of years after monitoring began.
    1. Show that \(k = 1.8\)
    2. Hence calculate the long-term population of goats predicted by this model. In a second model, the number of goats, \(x\), in thousands, is modelled by the differential equation $$3 \frac { \mathrm {~d} x } { \mathrm {~d} t } = x ( 9 - 2 x )$$
    3. Write \(\frac { 3 } { x ( 9 - 2 x ) }\) in partial fraction form.
    4. Solve the differential equation with the initial condition to show that $$x = \frac { 9 } { 2 + \mathrm { e } ^ { - 3 t } }$$
    5. Find the long-term population of goats predicted by this second model.
    Edexcel P4 2023 October Q8
    14 marks Standard +0.3
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7f5fc83d-ab7c-4edb-a2c6-7a58f1357d5a-24_579_642_251_715} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 6 t - 3 \sin 2 t \quad y = 2 \cos t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The curve meets the \(y\)-axis at 2 and the \(x\)-axis at \(k\), where \(k\) is a constant.
    1. State the value of \(k\).
    2. Use parametric differentiation to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \operatorname { cosec } t$$ where \(\lambda\) is a constant to be found. The point \(P\) with parameter \(\mathrm { t } = \frac { \pi } { 4 }\) lies on \(C\).
      The tangent to \(C\) at the point \(P\) cuts the \(y\)-axis at the point \(N\).
    3. Find the exact \(y\) coordinate of \(N\), giving your answer in simplest form. The region bounded by the curve, the \(x\)-axis and the \(y\)-axis is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
      1. Show that the volume of this solid is given by $$\int _ { 0 } ^ { \alpha } \beta ( 1 - \cos 4 t ) d t$$ where \(\alpha\) and \(\beta\) are constants to be found.
      2. Hence, using algebraic integration, find the exact volume of this solid.
    Edexcel P4 2018 Specimen Q1
    6 marks Moderate -0.3
    1. Use the binomial series to find the expansion of
    $$\frac { 1 } { ( 2 + 5 x ) ^ { 3 } } \quad | \boldsymbol { x } | < \frac { 2 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\) Give each coefficient as a fraction in its simplest form.
    VIIIV SIHI NI JIIIM ION OCVIIV SIHI NI JAHAM ION OOVJ4V SIHIL NI JIIIM IONOO
    Edexcel P4 2018 Specimen Q2
    7 marks Standard +0.3
    2. A curve \(C\) has the equation $$x ^ { 3 } + 2 x y - x - y ^ { 3 } - 20 = 0$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
    2. Find an equation of the tangent to \(C\) at the point \(( 3 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
      VIII SIHI NI I IIIM I O N OCVIIN SIHI NI JIHMM ION OOVI4V SIHI NI JIIYM ION OO
    Edexcel P4 2018 Specimen Q3
    10 marks Standard +0.3
    3. $$\mathrm { f } ( x ) = \frac { 1 } { x ( 3 x - 1 ) ^ { 2 } } = \frac { A } { x } + \frac { B } { ( 3 x - 1 ) } + \frac { C } { ( 3 x - 1 ) ^ { 2 } }$$
    1. Find the values of the constants \(A , B\) and \(C\)
      1. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
      2. Find \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants.
        (6)