- In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
- Use the substitution \(u = \mathrm { e } ^ { x } - 3\) to show that
$$\int _ { \ln 5 } ^ { \ln 7 } \frac { 4 \mathrm { e } ^ { 3 x } } { \mathrm { e } ^ { x } - 3 } \mathrm {~d} x = a + b \ln 2$$
where \(a\) and \(b\) are constants to be found.
- Show, by integration, that
$$\int 3 \mathrm { e } ^ { x } \cos 2 x \mathrm {~d} x = p \mathrm { e } ^ { x } \sin 2 x + q \mathrm { e } ^ { x } \cos 2 x + c$$
where \(p\) and \(q\) are constants to be found and \(c\) is an arbitrary constant.