8.
$$I = \int _ { 0 } ^ { 5 } \mathrm { e } ^ { \sqrt { } ( 3 x + 1 ) } \mathrm { d } x$$
- Given that \(y = \mathrm { e } ^ { \sqrt { } ( 3 x + 1 ) }\), complete the table with the values of \(y\) corresponding to \(x = 2\), 3 and 4.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 |
| \(y\) | \(\mathrm { e } ^ { 1 }\) | \(\mathrm { e } ^ { 2 }\) | | | | \(\mathrm { e } ^ { 4 }\) |
- Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the original integral \(I\), giving your answer to 4 significant figures.
- Use the substitution \(t = \sqrt { } ( 3 x + 1 )\) to show that \(I\) may be expressed as \(\int _ { a } ^ { b } k t e ^ { t } \mathrm {~d} t\), giving the values of \(a , b\) and \(k\).
- Use integration by parts to evaluate this integral, and hence find the value of \(I\) correct to 4 significant figures, showing all the steps in your working.