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.

Fig. 9 shows the curve \(y = f(x)\), where \(f(x) = e^{2x} + k e^{-2x}\) and \(k\) is a constant greater than 1. The curve crosses the \(y\)-axis at P and has a turning point Q. \includegraphics{figure_9}

1. Find the \(y\)-coordinate of P in terms of \(k\). [1]
2. Show that the \(x\)-coordinate of Q is \(\frac{1}{4}\ln k\), and find the \(y\)-coordinate in its simplest form. [5]
3. Find, in terms of \(k\), the area of the region enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{4}\ln k\). Give your answer in the form \(ak + b\). [4]

The function \(g(x)\) is defined by \(g(x) = f(x + \frac{1}{4}\ln k)\).

1. 1. Show that \(g(x) = \sqrt{k}(e^{2x} + e^{-2x})\). [3]
   2. Hence show that \(g(x)\) is an even function. [2]
   3. Deduce, with reasons, a geometrical property of the curve \(y = f(x)\). [3]

This is the graph of \(y = \frac{5}{4x-3} - \frac{3}{2}\) \includegraphics{figure_7} Find the area between the graph, the \(x\) axis, and the lines \(x = 1\) and \(x = 7\) in the form \(a \ln b + c\) where \(a, b, c \in Q\) [6]

This is the graph of \(y = \frac{5}{4x-3} - \frac{3}{2}\) \includegraphics{figure_6} Find the area between the graph, the \(x\) axis, and the lines \(x = 1\) and \(x = 7\) in the form \(a \ln b + c\) where \(a,b,c \in Q\) [6]

Find the exact value of \(\int_0^1 (e^x - x) dx\). [4]