1.08h Integration by substitution

16 In this question you must show detailed reasoning.
Find \(\int \frac { x } { 1 + \sqrt { x } } d x\). END OF QUESTION PAPER

8

1. The curve \(y = \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }\) is shown in Fig. 8. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-08_495_1058_1105_315} \captionsetup{labelformat=empty} \caption{Fig. 8}
   \end{figure}
   1. Show that \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 20 x ^ { 2 } - 4 } { \left( 1 + x ^ { 2 } \right) ^ { 4 } }\).
   2. In this question you must show detailed reasoning. Find the set of values of \(x\) for which the curve is concave downwards.
2. Use the substitution \(x = \tan \theta\) to find the exact value of \(\int _ { - 1 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } d x\). Answer all the questions.
   Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

4

1. Use integration by parts to find \(\int x \sin x \mathrm {~d} x\).
2. Using the substitution \(u = x ^ { 2 } + 5\), or otherwise, find \(\int x \sqrt { x ^ { 2 } + 5 } \mathrm {~d} x\).
3. The diagram shows the curve \(y = x ^ { 2 } - 9\) for \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-3_844_663_685_694} The shaded region \(R\) is bounded by the curve, the lines \(y = 1\) and \(y = 2\), and the \(y\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-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 \(x\).

6

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 0.4 } \cos \sqrt { 3 x + 1 } \mathrm {~d} x\), giving your answer to three significant figures.
2. Use the substitution \(u = 3 x + 1\) to find the exact value of \(\int _ { 0 } ^ { 1 } x \sqrt { 3 x + 1 } \mathrm {~d} x\).
   (6 marks)

6

1. Given that \(x = \frac { 1 } { \sin \theta }\), use the quotient rule to show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta\).
   (3 marks)
2. Use the substitution \(x = \operatorname { cosec } \theta\) to find \(\int _ { \sqrt { 2 } } ^ { 2 } \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x\), giving your answer to three significant figures.
   (9 marks)

8

1. Show that $$\int _ { 0 } ^ { \ln 2 } \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x = \frac { 3 } { 8 } \mathrm { e }$$
2. Use the substitution \(u = \tan x\) to find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec ^ { 4 } x \sqrt { \tan x } d x$$ (8 marks)

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x \sin 2 x\).
2. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 2 } - 6 \right) ^ { 4 }\).
   2. Hence, or otherwise, find \(\int x \left( x ^ { 2 } - 6 \right) ^ { 3 } \mathrm {~d} x\).

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 \(\int \frac { 1 + \ln x } { x } \mathrm {~d} x\).

2

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = ( 3 x - 1 ) ^ { 10 }\).
2. Use the substitution \(u = 2 x + 1\) to find \(\int x ( 2 x + 1 ) ^ { 8 } \mathrm {~d} x\), giving your answer in terms of \(x\).

7

1. Given that \(y = \frac { \sin \theta } { \cos \theta }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec ^ { 2 } \theta\).
2. Given that \(x = \sin \theta\), show that \(\frac { x } { \sqrt { 1 - x ^ { 2 } } } = \tan \theta\).
3. Use the substitution \(x = \sin \theta\) to find \(\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\), giving your answer in terms of \(x\).

7

1. Use integration by parts to find \(\int ( t - 1 ) \ln t \mathrm {~d} t\).
2. Use the substitution \(t = 2 x + 1\) to show that \(\int 4 x \ln ( 2 x + 1 ) \mathrm { d } x\) can be written as \(\int ( t - 1 ) \ln t \mathrm {~d} t\).
3. Hence find the exact value of \(\int _ { 0 } ^ { 1 } 4 x \ln ( 2 x + 1 ) \mathrm { d } x\).

4

1. Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to \(\int _ { 0.5 } ^ { 2 } \frac { x } { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your answer to three significant figures.
2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x ^ { 3 } } \mathrm {~d} x\).

6 Use the substitution \(u = x ^ { 4 } + 2\) to find the value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 7 } } { \left( x ^ { 4 } + 2 \right) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(p \ln q + r\), where \(p , q\) and \(r\) are rational numbers.

7 Use the substitution \(u = 3 - x ^ { 3 }\) to find the exact value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { 3 - x ^ { 3 } } \mathrm {~d} x\).
[0pt] [6 marks]

8 Use the substitution \(u = 4 x - 1\) to find the exact value of $$\int _ { \frac { 1 } { 4 } } ^ { \frac { 1 } { 2 } } ( 5 - 2 x ) ( 4 x - 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x$$

5

1. Find \(\int x \sqrt { x ^ { 2 } + 3 } \mathrm {~d} x\).
   (2 marks)
2. Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x \sqrt { x ^ { 2 } + 3 } } { \mathrm { e } ^ { 2 y } }$$ given that \(y = 0\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\).

2

1. Given that \(\frac { 4 x ^ { 3 } - 2 x ^ { 2 } + 16 x - 3 } { 2 x ^ { 2 } - x + 2 }\) can be expressed as \(A x + \frac { B ( 4 x - 1 ) } { 2 x ^ { 2 } - x + 2 }\), find the values of the constants \(A\) and \(B\).
2. The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 2 x ^ { 2 } + 16 x - 3 } { 2 x ^ { 2 } - x + 2 }$$ The point \(( - 1,2 )\) lies on the curve. Find the equation of the curve.
   [0pt] [4 marks]

3. Use the substitution \(x = \sin \theta\) to show that, for \(| x | \leq 1\), $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { x } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } + c \text {, where } c \text { is an arbitrary constant. }$$

6. (a) Find $$\int 2 \sin 3 x \sin 2 x d x$$ (b) Use the substitution \(u ^ { 2 } = x + 1\) to evaluate $$\int _ { 0 } ^ { 3 } \frac { x ^ { 2 } } { \sqrt { x + 1 } } \mathrm {~d} x$$ 6. continued

7. \begin{figure}[h] \end{figure} Figure 2 shows the curve with parametric equations $$x = \cos 2 t , \quad y = \operatorname { cosec } t , \quad 0 < t < \frac { \pi } { 2 } .$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac { 1 } { 2 }\).

1. Find the value of the parameter \(t\) at \(P\).
2. Show that the tangent to the curve at \(P\) has the equation $$y = 2 x + 1$$ The shaded region is bounded by the curve, the coordinate axes and the line \(x = \frac { 1 } { 2 }\).
3. Show that the area of the shaded region is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } k \cos t \mathrm {~d} t$$ where \(k\) is a positive integer to be found.
4. Hence find the exact area of the shaded region.
   7. continued
   7. continued

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve with parametric equations $$x = 3 \sin t , \quad y = 2 \sin 2 t , \quad 0 \leq t < \pi .$$ The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\).

1. Find the value of \(t\) at \(O\) and the value of \(t\) at \(A\). The region enclosed by the curve is rotated through \(\pi\) radians about the \(x\)-axis.
2. Show that the volume of the solid formed is given by $$\int _ { 0 } ^ { \frac { \pi } { 2 } } 12 \pi \sin ^ { 2 } 2 t \cos t \mathrm {~d} t$$
3. Using the substitution \(u = \sin t\), or otherwise, evaluate this integral, giving your answer as an exact multiple of \(\pi\).
   6. continued

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve with parametric equations $$x = 2 - t ^ { 2 } , \quad y = t ( t + 1 ) , \quad t \geq 0$$

1. Find the coordinates of the points where the curve meets the coordinate axes.
2. Find the exact area of the region bounded by the curve and the coordinate axes.
   6. continued

8. (a) Show that the substitution \(u = \sin x\) transforms the integral $$\int \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ into the integral $$\int \frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) } \mathrm { d } u .$$ (b) Express \(\frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) }\) in partial fractions.
(c) Hence, evaluate $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ giving your answer in the form \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are integers.
8. continued
8. continued

6. (a) Use the substitution \(x = 2 \sin u\) to evaluate $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$$ (b) Use integration by parts to evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x d x$$ 6. continued