- In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.}
- Use the substitution \(t = \tan \left( \frac { \theta } { 2 } \right)\) to show that
$$\int \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \int \frac { a } { ( t + b ) ^ { 2 } + c } d t$$
where \(a\), \(b\) and \(c\) are constants to be determined.
- Hence show that
$$\int _ { \frac { \pi } { 2 } } ^ { \frac { 2 \pi } { 3 } } \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \ln \left( \frac { 2 \sqrt { 3 } } { 3 } \right)$$