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

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 $$\int _ { x } ^ { x + 1 } \frac { \ln t } { 2 ^ { t } } \mathrm {~d} t$$

11

1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
2. Find
   1. \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
   2. \(\quad \int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).

12

1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
2. Find
   1. \(\quad \int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
   2. \(\quad \int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).

12

1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
2. Find
   1. \(\quad \int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
   2. \(\quad \int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).

12

1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
2. Find
   1. \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
   2. \(\int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = xe^{-\frac{1}{2}x}\), \(x > 0\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the line \(x = 4\). The table shows corresponding values of \(x\) and \(y\) for \(y = xe^{-\frac{1}{2}x}\).

1. Complete the table with the value of \(y\) corresponding to \(x = 2\) [1]
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places. [4]
3. 1. Find \(\int xe^{-\frac{1}{2}x} \, dx\).
   2. Hence find the exact area of \(R\), giving your answer in the form \(a + be^{-2}\), where \(a\) and \(b\) are integers. [6]

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = 4x - xe^{\frac{1}{x}}, x \geqslant 0\) The curve meets the \(x\)-axis at the origin \(O\) and cuts the \(x\)-axis at the point \(A\).

1. Find, in terms of \(\ln 2\), the \(x\) coordinate of the point \(A\). [2]
2. Find $$\int xe^{\frac{1}{x}} dx$$ [3]
3. Find, by integration, the exact value for the area of \(R\). Give your answer in terms of \(\ln 2\) [3]

The finite region \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis and the curve with equation $$y = 4x - xe^{\frac{1}{x}}, x \geqslant 0$$

Fig. 8 shows the line \(y = x\) and parts of the curves \(y = f(x)\) and \(y = g(x)\), where $$f(x) = e^{x-1}, \quad g(x) = 1 + \ln x.$$ The curves intersect the axes at the points A and B, as shown. The curves and the line \(y = x\) meet at the point C. \includegraphics{figure_8}

1. Find the exact coordinates of A and B. Verify that the coordinates of C are \((1, 1)\). [5]
2. Prove algebraically that \(g(x)\) is the inverse of \(f(x)\). [2]
3. Evaluate \(\int_0^1 f(x) \, dx\), giving your answer in terms of \(e\). [3]
4. Use integration by parts to find \(\int \ln x \, dx\). Hence show that \(\int_{e^{-1}}^1 g(x) \, dx = \frac{1}{e}\). [6]
5. Find the area of the region enclosed by the lines OA and OB, and the arcs AC and BC. [2]

A curve is defined by the equation \(y = 2x \ln(1 + x)\).

1. Find \(\frac{dy}{dx}\) and hence verify that the origin is a stationary point of the curve. [4]
2. Find \(\frac{d^2y}{dx^2}\) and use this to verify that the origin is a minimum point. [5]
3. Using the substitution \(u = 1 + x\), show that \(\int \frac{x^2}{1+x} \, dx = \int \left(u - 2 + \frac{1}{u}\right) du\). Hence evaluate \(\int_0^1 \frac{x^2}{1+x} \, dx\), giving your answer in an exact form. [6]
4. Using integration by parts and your answer to part (iii), evaluate \(\int_0^1 2x \ln(1 + x) \, dx\). [4]