Sequential multi-part (building on previous)

8. (i) Find \(\int x \sin x d x\) (ii) (a) Use the substitution \(x = \sec \theta\) to show that
(b) Hence find the exact value of $$\int _ { 1 } ^ { 2 } \sqrt { 1 - \frac { 1 } { x ^ { 2 } } } \mathrm {~d} x = \int _ { 0 } ^ { \frac { \pi } { 3 } } \tan ^ { 2 } \theta \mathrm {~d} \theta$$ Hence find the exact value of $$\int _ { 1 } ^ { 2 } \sqrt { 1 - \frac { 1 } { x ^ { 2 } } } \mathrm {~d} x$$

2. (a) Use integration by parts to find \(\int x \sin 3 x \mathrm {~d} x\).
(b) Using your answer to part (a), find \(\int x ^ { 2 } \cos 3 x \mathrm {~d} x\).

3. (a) Find \(\int x \cos 2 x d x\).
(b) Hence, using the identity \(\cos 2 x = 2 \cos ^ { 2 } x - 1\), deduce \(\int x \cos ^ { 2 } x \mathrm {~d} x\).

2. (a) Use integration by parts to find \(\int x \mathrm { e } ^ { x } \mathrm {~d} x\).
(b) Hence find \(\int x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).

6. (a) Find \(\int \sqrt { } ( 5 - x ) \mathrm { d } x\).
(2) \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with equation $$y = ( x - 1 ) \sqrt { } ( 5 - x ) , \quad 1 \leqslant x \leqslant 5$$ (b) (i) Using integration by parts, or otherwise, find $$\int ( x - 1 ) \sqrt { } ( 5 - x ) \mathrm { d } x$$ (ii) Hence find \(\int _ { 1 } ^ { 5 } ( x - 1 ) \sqrt { } ( 5 - x ) \mathrm { d } x\).

4

1. Use integration by parts to find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
2. Hence find \(\int x \tan ^ { 2 } x \mathrm {~d} x\).

8 \begin{figure}[h] \end{figure} Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } + 2 }\).

1. Show algebraically that \(\mathrm { f } ( x )\) is an even function, and state how this property relates to the curve \(y = \mathrm { f } ( x )\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\).
3. Show that \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } }\).
4. Hence, using the substitution \(u = \mathrm { e } ^ { x } + 1\), or otherwise, find the exact area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, and the lines \(x = 0\) and \(x = 1\).
5. Show that there is only one point of intersection of the curves \(y = \mathrm { f } ( x )\) and \(y = \frac { 1 } { 4 } \mathrm { e } ^ { x }\), and find its coordinates.

8

1. Use division to show that \(\frac { t ^ { 3 } } { t + 2 } \equiv t ^ { 2 } - 2 t + 4 - \frac { 8 } { t + 2 }\).
2. Find \(\int _ { 1 } ^ { 2 } 6 t ^ { 2 } \ln ( t + 2 ) \mathrm { d } t\). Give your answer in the form \(A + B \ln 3 + C \ln 4\).

7

1. Use integration by parts to find:
   1. \(\quad \int x \cos 4 x \mathrm {~d} x\);
      (4 marks)
   2. \(\int x ^ { 2 } \sin 4 x d x\).
      (4 marks)
2. The region bounded by the curve \(y = 8 x \sqrt { ( \sin 4 x ) }\) and the lines \(x = 0\) and \(x = 0.2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the value of the volume of the solid generated, giving your answer to three significant figures.
   (3 marks)

10

1. 1. By writing \(\ln x\) as \(( \ln x ) \times 1\), use integration by parts to find \(\int \ln x \mathrm {~d} x\).
   2. Find \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\).
2. Use the substitution \(u = \sqrt { x }\) to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 1 } { x + \sqrt { x } } \mathrm {~d} x$$ (7 marks)

6. (a) Use the substitution \(u = \sqrt { t }\) to show that $$\int _ { 1 } ^ { x } \frac { \ln t } { \sqrt { t } } \mathrm {~d} t = 4 - 4 \sqrt { x } + 2 \sqrt { x } \ln x \quad x \geqslant 1$$ (b) The function g is such that $$\int _ { 1 } ^ { x } \mathrm {~g} ( t ) \mathrm { d } t = x - \sqrt { x } \ln x - 1 \quad x \geqslant 1$$

1. Use differentiation to find the function g .
2. Evaluate \(\int _ { 4 } ^ { 16 } \mathrm {~g} ( t ) \mathrm { d } t\) and simplify your answer.
   (c) Find the value of \(x\) (where \(x > 1\) ) that gives the maximum value of $$\int _ { x } ^ { x + 1 } \frac { \ln t } { 2 ^ { t } } \mathrm {~d} t$$