Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

(i) Use the substitution $u=e^x-3$ to show that

$$\int_{\ln 5}^{\ln 7}\frac{4e^{3x}}{e^x-3}\,dx=a+b\ln 2$$

where $a$ and $b$ are constants to be found.

(ii) Show, by integration, that

$$\int 3e^x\cos 2x\,dx=pe^x\sin 2x+qe^x\cos 2x+c$$

where $p$ and $q$ are constants to be found and $c$ is an arbitrary constant.