Definite integral with exponentials

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1. Find the exact value of \(\int _ { 0 } ^ { 2 } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x\).
2. Find \(\int \left( \tan ^ { 2 } x + 4 \sin ^ { 2 } 2 x \right) \mathrm { d } x\).

1 Find the exact value of \(\int _ { - 1 } ^ { 2 } \left( 4 \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { - x } \right) \mathrm { d } x\).

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1. Find the value of \(\int _ { 0 } ^ { \frac { 2 } { 3 } \pi } \sin \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).
2. Find \(\int \mathrm { e } ^ { - x } \left( 1 + \mathrm { e } ^ { x } \right) \mathrm { d } x\).

3 Without using a calculator, find the exact value of \(\int _ { 0 } ^ { 2 } 4 \mathrm { e } ^ { - x } \left( \mathrm { e } ^ { 3 x } + 1 \right) \mathrm { d } x\).

3 The definite integral \(I\) is defined by \(I = \int _ { 0 } ^ { 2 } \left( 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3 \right) \mathrm { d } x\).

1. Show that \(I = 8 \mathrm { e } - 2\).
2. Sketch the curve \(y = 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3\) for \(0 \leqslant x \leqslant 2\).
3. State whether an estimate of \(I\) obtained by using the trapezium rule will be more than or less than \(8 \mathrm { e } - 2\). Justify your answer.

6. Find the value of each of the following integrals in exact, simplified form.

1. \(\quad \int _ { - 1 } ^ { 0 } \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\)
2. \(\int _ { 2 } ^ { 4 } \frac { 3 x ^ { 2 } - 2 } { x } \mathrm {~d} x\)

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\includegraphics[max width=\textwidth, alt={}, center]{6d15cb4d-f540-488b-b94e-7a494f192ba5-2_469_721_1932_662} The diagram shows the curves \(y = \mathrm { e } ^ { 2 x }\) and \(y = 8 \mathrm { e } ^ { - x }\). The shaded region is bounded by the curves and the \(y\)-axis. Without using a calculator,

1. solve an appropriate equation to show that the curves intersect at a point for which \(x = \ln 2\),
2. find the area of the shaded region, giving your answer in simplified form.