6
3
\end{array} \right) + \mu \left( \begin{array} { r }
1
0
- 2
\end{array} \right)$$
Find the acute angle between the lines.
6 Two students are trying to evaluate the integral \(\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x\).
Sarah uses the trapezium rule with 2 strips, and starts by constructing the following table.
| \(x\) | 1 | 1.5 | 2 |
| \(\sqrt { 1 + \mathrm { e } ^ { - x } }\) | 1.1696 | 1.1060 | 1.0655 |
- Complete the calculation, giving your answer to 3 significant figures.
Anish uses a binomial approximation for \(\sqrt { 1 + \mathrm { e } ^ { - x } }\) and then integrates this.
- Show that, provided \(\mathrm { e } ^ { - x }\) is suitably small, \(\left( 1 + \mathrm { e } ^ { - x } \right) ^ { \frac { 1 } { 2 } } \approx 1 + \frac { 1 } { 2 } \mathrm { e } ^ { - x } - \frac { 1 } { 8 } \mathrm { e } ^ { - 2 x }\).
- Use this result to evaluate \(\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x\) approximately, giving your answer to 3 significant figures.