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

1. Given that \(A\) is constant and

$$\int _ { 1 } ^ { 4 } ( 3 \sqrt { x } + A ) \mathrm { d } x = 2 A ^ { 2 }$$ show that there are exactly two possible values for \(A\).

7 The diagram shows part of a curve which passes through the point \(( 1,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-4_711_704_1722_258} The gradient of the curve is given by \(\frac { d y } { d x } = 6 x + \frac { 8 } { x ^ { 3 } }\).
Determine whether the curve passes through the point \(( 2,12 )\).

8

1. Show that \(8 \sin ^ { 2 } x \cos ^ { 2 } x\) can be written as \(1 - \cos 4 x\).
2. Hence find \(\int \sin ^ { 2 } x \cos ^ { 2 } x \mathrm {~d} x\).

9 The gradient of a curve is given by \(\frac { d y } { d x } = e ^ { x } - 4 e ^ { - x }\).

1. Show that the \(x\)-coordinate of any point on the curve at which the gradient is 3 satisfies the equation \(\left( e ^ { x } \right) ^ { 2 } - 3 e ^ { x } - 4 = 0\).
2. Hence show that there is only one point on the curve at which the gradient is 3 , stating the exact value of its \(x\)-coordinate.
3. The curve passes through the point \(( 0,0 )\). Show that when \(x = 1\) the curve is below the \(x\)-axis.

5 A particle \(P\) of mass 3 kg moves on the \(x\)-axis under the action of a single force acting in the positive \(x\)-direction. At time \(t \mathrm {~s}\), where \(t \geqslant 0\), the displacement of \(P\) is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the force acting is inversely proportional to \(( t + 1 ) ^ { 2 }\). Initially \(P\) is at rest at the point where \(x = 1\). When \(t = 1 , v = 2\).

1. Show that \(\frac { \mathrm { dv } } { \mathrm { dt } } = \frac { \mathrm { k } } { 3 ( \mathrm { t } + 1 ) ^ { 2 } }\) where \(k\) is a constant.
2. Find an expression for \(v\) in terms of \(t\).
3. Find an expression for \(x\) in terms of \(t\). As \(t\) increases, \(v\) approaches a limiting value, \(\mathrm { V } _ { \mathrm { T } }\).
4. Determine how far \(P\) is from its initial position at the instant when \(v\) is \(95 \%\) of \(\mathrm { V } _ { \mathrm { T } }\).

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 \(\int _ { 0 } ^ { 2 } x ^ { 5 } \sqrt { x ^ { 3 } + 1 } \mathrm {~d} x\).

3 For integers \(n \geqslant 0 , \mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } \frac { \mathrm { x } ^ { \mathrm { n } } } { 1 + \mathrm { x } ^ { 2 } } \mathrm { dx }\).

1. For integers \(n \geqslant 2\), show that \(I _ { n } + I _ { n - 2 } = \frac { 1 } { n - 1 }\).
2. 1. Determine the exact value of \(I _ { 10 }\).
   2. Deduce that \(\pi < 3 \frac { 107 } { 315 }\).

7 Let \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { \mathrm { n } } \mathrm { xdx }\) for integers \(n \geqslant 0\).

1. Show that, for \(n \geqslant 2 , \quad \mathrm { nl } _ { \mathrm { n } } = ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } - 2 }\).
2. Use this reduction formula to deduce the exact value of \(I _ { 8 }\).
3. Use the results of parts (a) and (b) to determine the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 6 } x \sin ^ { 2 } x d x\).

5 In this question you must show detailed reasoning.
It is given that \(I _ { n } = \int _ { 0 } ^ { \pi } \sin ^ { n } \theta \mathrm {~d} \theta\) for \(n \geq 0\).

1. Prove that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geq 2\).
2. (a) Evaluate \(I _ { 1 }\).
   (b) Use the reduction formula to determine the exact value of \(\int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta \sin ^ { 5 } \theta \mathrm {~d} \theta\).

9 The sketch shows the graph of \(y = 4 - \mathrm { e } ^ { 2 x }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\). \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-5_711_921_466_557}

1. 1. Find \(\int \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
      (2 marks)
   2. Hence show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x = 4 \ln 2 - \frac { 3 } { 2 }\).
2. 1. Write down the \(y\)-coordinate of \(A\).
   2. Show that \(x = \ln 2\) at \(B\).
3. Find the equation of the normal to the curve \(y = 4 - \mathrm { e } ^ { 2 x }\) at the point \(B\).
4. Find the area of the region enclosed by the curve \(y = 4 - \mathrm { e } ^ { 2 x }\), the normal to the curve at \(B\) and the \(y\)-axis.

5

1. 1. Given that \(y = 2 x ^ { 2 } - 8 x + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence, or otherwise, find $$\int _ { 4 } ^ { 6 } \frac { x - 2 } { 2 x ^ { 2 } - 8 x + 3 } d x$$ giving your answer in the form \(k \ln 3\), where \(k\) is a rational number.
2. Use the substitution \(u = 3 x - 1\) to find \(\int x \sqrt { 3 x - 1 } \mathrm {~d} x\), giving your answer in terms of \(x\).

5

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

3

1. Given that \(y = 4 x ^ { 3 } - 6 x + 1\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (l mark)
2. Hence find \(\int _ { 2 } ^ { 3 } \frac { 2 x ^ { 2 } - 1 } { 4 x ^ { 3 } - 6 x + 1 } \mathrm {~d} x\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.

2

1. Use Simpson's rule, with five ordinates (four strips), to calculate an estimate for $$\int _ { 0 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x$$ Give your answer to four significant figures.
2. Show that the exact value of \(\int _ { 0 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x\) is \(\ln k\), where \(k\) is an integer. (5 marks)

3

1. Find \(\int \mathrm { e } ^ { 4 x } \mathrm {~d} x\).
2. Use integration by parts to find \(\int \mathrm { e } ^ { 4 x } ( 2 x + 1 ) \mathrm { d } x\).
3. By using the substitution \(u = 1 + \ln x\), or otherwise, find \(\int \frac { 1 + \ln x } { x } \mathrm {~d} x\).

4

1. A curve is defined by the parametric equations \(x = 3 \cos 2 \theta , y = 2 \cos \theta\).
   1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { k \cos \theta }\), where \(k\) is an integer.
   2. Find an equation of the normal to the curve at the point where \(\theta = \frac { \pi } { 3 }\).
2. Find the exact value of \(\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \sin ^ { 2 } x \mathrm {~d} x\).

1 It is given that \(\mathrm { f } ( x ) = \frac { 19 x - 2 } { ( 5 - x ) ( 1 + 6 x ) }\) can be expressed as \(\frac { A } { 5 - x } + \frac { B } { 1 + 6 x }\), where \(A\) and \(B\) are integers.

1. Find the values of \(A\) and \(B\).
2. Hence show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = k \ln 5\), where \(k\) is a rational number.
   [0pt] [6 marks]

1. Find

$$\int \cot ^ { 2 } 2 x \mathrm {~d} x$$

3 A particle moves in a straight line and at time \(t\) has velocity \(v\), where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. State the range of values of the acceleration of the particle.
2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

2 A particle moves in a straight line and at time \(t\) it has velocity \(v\), where $$v = 3 t ^ { 2 } - 2 \sin 3 t + 6$$

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. When \(t = \frac { \pi } { 3 }\), show that the acceleration of the particle is \(2 \pi + 6\).
2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

3 The region \(R\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = a \mathrm { e } ^ { \frac { x } { a } }\) and the line \(x = a \ln 2\) (where \(a\) is a positive constant). A uniform solid of revolution, of mass \(M\), is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Find, in terms of \(M\) and \(a\), the moment of inertia about the \(x\)-axis of this solid of revolution.
[0pt] [8]

3 The region bounded by the \(y\)-axis and the curves \(y = \sin 2 x\) and \(y = \sqrt { 2 } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) is occupied by a uniform lamina. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina.

8

1. 1. Expand $$\left( z + \frac { 1 } { z } \right) \left( z - \frac { 1 } { z } \right)$$
   2. Hence, or otherwise, expand $$\left( z + \frac { 1 } { z } \right) ^ { 4 } \left( z - \frac { 1 } { z } \right) ^ { 2 }$$
2. 1. Use De Moivre's theorem to show that if \(z = \cos \theta + \mathrm { i } \sin \theta\) then $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
   2. Write down a corresponding result for \(z ^ { n } - \frac { 1 } { z ^ { n } }\).
3. Hence express \(\cos ^ { 4 } \theta \sin ^ { 2 } \theta\) in the form $$A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$ where \(A , B , C\) and \(D\) are rational numbers.
4. Find \(\int \cos ^ { 4 } \theta \sin ^ { 2 } \theta d \theta\).

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

$$I _ { n } = \int \frac { \cos ( n x ) } { \sin x } \mathrm {~d} x \quad n \geqslant 1$$

1. Show that, for \(n \geqslant 1\) $$I _ { n + 2 } = \frac { 2 \cos ( n + 1 ) x } { n + 1 } + I _ { n }$$
2. Hence determine the exact value of $$\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \frac { \cos ( 5 x ) } { \sin x } d x$$ giving the answer in the form \(a + b \ln c\) where \(a , b\) and \(c\) are rational numbers to be found.