Exponential and logarithmic integration

4. (i) Use Simpson's rule with four intervals, each of width 0.25 , to estimate the value of the integral $$\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ (ii) Find the exact value of the integral $$\int _ { \frac { 1 } { 2 } } ^ { 1 } e ^ { 1 - 2 x } d x$$

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.

8 The diagram shows the curves \(y = \mathrm { e } ^ { 2 x } - 1\) and \(y = 4 \mathrm { e } ^ { - 2 x } + 2\). \includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-6_958_1492_372_242} The curve \(y = 4 \mathrm { e } ^ { - 2 x } + 2\) crosses the \(y\)-axis at the point \(A\) and the curves intersect at the point \(B\).

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x } - 1\).
2. Write down the coordinates of the point \(A\).
3. 1. Show that the \(x\)-coordinate of the point \(B\) satisfies the equation $$\left( \mathrm { e } ^ { 2 x } \right) ^ { 2 } - 3 \mathrm { e } ^ { 2 x } - 4 = 0$$
   2. Hence find the exact value of the \(x\)-coordinate of the point \(B\).
4. Find the exact value of the area of the shaded region bounded by the curves \(y = \mathrm { e } ^ { 2 x } - 1\) and \(y = 4 \mathrm { e } ^ { - 2 x } + 2\) and the \(y\)-axis.

5. \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). Figure 1 shows the part of \(C\) for which \(0 \leq x \leq 2\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { x } - 2 x ^ { 2 } ,$$ and that \(C\) has a single maximum, at \(x = k\),

1. show that \(1.48 < k < 1.49\). Given also that the point \(( 0,5 )\) lies on \(C\),
2. find \(\mathrm { f } ( x )\). The finite region \(R\) is bounded by \(C\), the coordinate axes and the line \(x = 2\).
3. Use integration to find the exact area of \(R\).
   (4)