6.
$$\mathrm { f } ( x ) = \frac { x + 2 } { x ^ { 2 } + 9 }$$
- Show that
$$\int \mathrm { f } ( x ) \mathrm { d } x = A \ln \left( x ^ { 2 } + 9 \right) + B \arctan \left( \frac { x } { 3 } \right) + c$$
where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
- Hence show that the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\) is
$$\frac { 1 } { 6 } \ln 2 + \frac { 1 } { 18 } \pi$$
- Use the answer to part (b) to find the mean value, over the interval \([ 0,3 ]\), of
$$\mathrm { f } ( x ) + \ln k$$
where \(k\) is a positive constant, giving your answer in the form \(p + \frac { 1 } { 6 } \ln q\), where \(p\) and \(q\) are constants and \(q\) is in terms of \(k\).