- In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
$$I _ { n } = \int \frac { \cos ( n x ) } { \sin x } \mathrm {~d} x \quad n \geqslant 1$$
- Show that, for \(n \geqslant 1\)
$$I _ { n + 2 } = \frac { 2 \cos ( n + 1 ) x } { n + 1 } + I _ { n }$$
- Hence determine the exact value of
$$\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \frac { \cos ( 5 x ) } { \sin x } d x$$
giving the answer in the form \(a + b \ln c\) where \(a , b\) and \(c\) are rational numbers to be found.