1.08h Integration by substitution

11 \includegraphics[max width=\textwidth, alt={}, center]{7cdf4db7-7217-4ef1-becf-359a70cfeb62-16_556_698_274_712} The diagram shows the curve \(y = \sin x \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 significant figures.
2. Using the substitution \(u = \cos x\), find the area of the shaded region enclosed by the curve and the \(x\)-axis in the first quadrant, giving your answer in a simplified exact form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 Using the substitution \(u = \sqrt { x }\), find the exact value of $$\int _ { 3 } ^ { \infty } \frac { 1 } { ( x + 1 ) \sqrt { x } } \mathrm {~d} x$$

8 \includegraphics[max width=\textwidth, alt={}, center]{3c63c42a-2658-4984-93e8-b2a8d18eb37a-12_473_839_274_644} The diagram shows part of the curve \(y = \sin \sqrt { x }\). This part of the curve intersects the \(x\)-axis at the point where \(x = a\).

1. State the exact value of \(a\).
2. Using the substitution \(u = \sqrt { x }\), find the exact area of the shaded region in the first quadrant bounded by this part of the curve and the \(x\)-axis.

9 \includegraphics[max width=\textwidth, alt={}, center]{ce3c4a9c-bf83-4d28-96e2-ef31c3673dea-12_375_645_274_742} The diagram shows the curve \(y = x \mathrm { e } ^ { - \frac { 1 } { 4 } x ^ { 2 } }\), for \(x \geqslant 0\), and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. Using the substitution \(x = \sqrt { u }\), or otherwise, find by integration the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 3\).

11 \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_565_634_260_717} The diagram shows the curve \(y = 2 \sin x \sqrt { 2 + \cos x }\), for \(0 \leqslant x \leqslant 2 \pi\), and its minimum point \(M\), where \(x = a\).

1. Find the value of \(a\) correct to 2 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_2716_38_109_2012} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-19_2726_33_97_22}
2. Use the substitution \(u = 2 + \cos x\) to find the exact area of the shaded region \(R\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

4 Find the solution of the differential equation $$\sin \theta \frac { d y } { d \theta } + y = \tan \frac { 1 } { 2 } \theta$$ where \(0 < \theta < \pi\), given that \(y = 1\) when \(\theta = \frac { 1 } { 2 } \pi\). Give your answer in the form \(y = \mathrm { f } ( \theta )\). [You may use without proof the result that \(\int \operatorname { cosec } \theta d \theta = \ln \tan \frac { 1 } { 2 } \theta\).]

7 The integral \(\mathrm { I } _ { \mathrm { n } }\), where n is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 3 } { 2 } } \left( 4 + \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { 1 }\), expressing your answer in logarithmic form.
2. By considering \(\frac { d } { d x } \left( x \left( 4 + x ^ { 2 } \right) ^ { - \frac { 1 } { 2 } n } \right)\), or otherwise, show that $$4 n l _ { n + 2 } = \frac { 3 } { 2 } \left( \frac { 2 } { 5 } \right) ^ { n } + ( n - 1 ) l _ { n } .$$
3. Find the value of \(I _ { 5 }\).

8

1. Find \(\int \sin \theta \cos ^ { n } \theta d \theta\), where \(n \neq - 1\).
   Let \(I _ { m , n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { m } \theta \cos ^ { n } \theta d \theta\).
2. Show that, for \(m \geqslant 2\) and \(n \geqslant 0\), $$I _ { m , n } = \frac { m - 1 } { m + n } I _ { m - 2 , n }$$
3. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 5 }\), where \(z = \cos \theta + i \sin \theta\), use de Moivre's theorem to show that $$\cos ^ { 5 } \theta = a \cos 5 \theta + b \cos 3 \theta + c \cos \theta$$ where \(a\), \(b\) and \(c\) are constants to be determined.
4. Using the results given in parts (b) and (c), find the exact value of \(I _ { 2,5 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

3

1. By considering the binomial expansion of \(\left( z + z ^ { - 1 } \right) ^ { 4 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that \(\cos ^ { 4 } \theta = \frac { 1 } { 8 } ( \cos 4 \theta + 4 \cos 2 \theta + 3 )\).
2. Use the substitution \(x = \sin \theta\) to find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } \mathrm {~d} x\).

4 The integral \(\mathrm { I } _ { \mathrm { n } }\) is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } \left( 1 + \mathrm { x } ^ { 5 } \right) ^ { \mathrm { n } } \mathrm { dx }\).

1. By considering \(\frac { d } { d x } \left( x \left( 1 + x ^ { 5 } \right) ^ { n } \right)\), or otherwise, show that $$( 5 n + 1 ) l _ { n } = 2 ^ { n } + 5 n l _ { n - 1 }$$
2. Find the exact value of \(I _ { 3 }\).

7

1. Use the substitution \(\mathrm { u } = \mathrm { x } ^ { 2 } - 1\) to find \(\int \frac { x } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x\). \includegraphics[max width=\textwidth, alt={}, center]{d3ddf5ce-4399-4438-ab67-7bdb2e1bea6e-12_778_1548_1007_296} The diagram shows the curve with equation \(\mathrm { y } = \cosh ^ { - 1 } \mathrm { x }\) together with a set of \(( N - 1 )\) rectangles of unit width.
2. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 2 } ^ { N } \ln \left( r + \sqrt { r ^ { 2 } - 1 } \right) > N \ln \left( N + \sqrt { N ^ { 2 } - 1 } \right) - \sqrt { N ^ { 2 } - 1 }$$
3. Use a similar method to find, in terms of \(N\), an upper bound for \(\sum _ { \mathrm { r } = 2 } ^ { \mathrm { N } } \ln \left( \mathrm { r } + \sqrt { \mathrm { r } ^ { 2 } - 1 } \right)\).

7

1. Use the substitution \(\mathrm { u } = \mathrm { x } ^ { 2 } - 1\) to find \(\int \frac { x } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x\). \includegraphics[max width=\textwidth, alt={}, center]{482b2236-1f1b-4c53-a1bc-0277cf63dc62-12_778_1548_1007_296} The diagram shows the curve with equation \(\mathrm { y } = \cosh ^ { - 1 } \mathrm { x }\) together with a set of \(( N - 1 )\) rectangles of unit width.
2. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 2 } ^ { N } \ln \left( r + \sqrt { r ^ { 2 } - 1 } \right) > N \ln \left( N + \sqrt { N ^ { 2 } - 1 } \right) - \sqrt { N ^ { 2 } - 1 }$$
3. Use a similar method to find, in terms of \(N\), an upper bound for \(\sum _ { \mathrm { r } = 2 } ^ { \mathrm { N } } \ln \left( \mathrm { r } + \sqrt { \mathrm { r } ^ { 2 } - 1 } \right)\).

7

1. Use the substitution \(\mathrm { u } = 1 + \mathrm { x } ^ { 2 }\) to find $$\int \frac { x } { \sqrt { 1 + x ^ { 2 } } } d x$$
2. Find the solution of the differential equation $$x \frac { d y } { d x } - y = x ^ { 2 } \sinh ^ { - 1 } x$$ given that \(y = 1\) when \(x = 1\). Give your answer in the form \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\).

1 Find the exact value of \(\int _ { 2 } ^ { \frac { 7 } { 2 } } \frac { 1 } { \sqrt { 4 x - x ^ { 2 } - 1 } } \mathrm {~d} x\).

3 A curve has equation \(y = \mathrm { e } ^ { x }\) for \(\ln \frac { 4 } { 3 } \leqslant x \leqslant \ln \frac { 12 } { 5 }\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).

1. Use the substitution \(u = \mathrm { e } ^ { x }\) to show that $$A = 2 \pi \int _ { \frac { 4 } { 3 } } ^ { \frac { 12 } { 5 } } \sqrt { 1 + u ^ { 2 } } \mathrm {~d} u$$
2. Use the substitution \(u = \sinh v\) to show that $$A = \pi \left( \frac { 904 } { 225 } + \ln \frac { 5 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-06_2716_38_109_2012} \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-07_2726_35_97_20}

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

3 A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\). One force has magnitude \(4 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the positive \(x\)-direction. The other force has magnitude \(2.4 x ^ { 2 } \mathrm {~N}\) and acts in the negative \(x\)-direction.

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \mathrm { e } ^ { - x } - 3 x ^ { 2 }\).
2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 2\).

6 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\) on a smooth horizontal surface. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A horizontal force of variable magnitude \(0.09 \sqrt { } x \mathrm {~N}\) directed away from \(O\) acts on \(P\). An additional force of constant magnitude 0.3 N directed towards \(O\) acts on \(P\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 0.45 \sqrt { } x - 1.5\).
2. Find the value of \(x\) for which the acceleration of \(P\) is zero.
3. Given that the minimum value of \(v\) is positive, find the set of possible values for the speed of projection.

5 A particle \(P\) of mass 0.5 kg is projected vertically upwards from a point on a horizontal surface. A resisting force of magnitude \(0.02 v ^ { 2 } \mathrm {~N}\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the upward velocity of \(P\) when it is a height of \(x \mathrm {~m}\) above the surface. The initial speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, while \(P\) is moving upwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 }\).
2. Find the greatest height of \(P\) above the surface.
3. Find the speed of \(P\) immediately before it strikes the surface after descending.

9. (i) Find $$\int \frac { ( 3 x + 2 ) ^ { 2 } } { 4 \sqrt { x } } \mathrm {~d} x \quad x > 0$$ giving your answer in simplest form.
(ii) A curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given

• \(\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } + a x + b\) where \(a\) and \(b\) are constants
• the \(y\) intercept of \(C\) is - 8
• the point \(P ( 3 , - 2 )\) lies on \(C\)
• the gradient of \(C\) at \(P\) is 2
  find, in simplest form, \(\mathrm { f } ( x )\).
  

\includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-31_2255_50_314_34}

6. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\) Given that

• \(\mathrm { f } ^ { \prime } ( x ) = \frac { ( x + 3 ) ^ { 2 } } { x \sqrt { x } }\)
• the point \(P ( 4,20 )\) lies on \(C\)
• Find \(\mathrm { f } ( x )\), simplifying your answer.
  

9. Find

1. \(\int \frac { 3 x - 2 } { 3 x ^ { 2 } - 4 x + 5 } \mathrm {~d} x\)
2. \(\int \frac { \mathrm { e } ^ { 2 x } } { \left( \mathrm { e } ^ { 2 x } - 1 \right) ^ { 3 } } \mathrm {~d} x \quad x \neq 0\)

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3. (i) Find, in simplest form, $$\int ( 2 x - 5 ) ^ { 7 } \mathrm {~d} x$$ (ii) Show, by algebraic integration, that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { 4 \sin x } { 1 + 2 \cos x } \mathrm {~d} x = \ln a$$ where \(a\) is a rational constant to be found.

3. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Given that \(k\) is a positive constant,

1. find $$\int \frac { 9 x } { 3 x ^ { 2 } + k } d x$$ Given also that $$\int _ { 2 } ^ { 5 } \frac { 9 x } { 3 x ^ { 2 } + k } \mathrm {~d} x = \ln 8$$
2. find the value of \(k\)

2. $$g ( x ) = \frac { 2 x ^ { 2 } - 5 x + 8 } { x - 2 }$$

1. Write \(g ( x )\) in the form $$A x + B + \frac { C } { x - 2 }$$ where \(A , B\) and \(C\) are integers to be found.
2. Hence use algebraic integration to show that $$\int _ { 4 } ^ { 8 } \mathrm {~g} ( x ) \mathrm { d } x = \alpha + \beta \ln 3$$ where \(\alpha\) and \(\beta\) are integers to be found.