3.
$$f ( x ) = \frac { 1 } { \sqrt { 4 x ^ { 2 } + 9 } }$$
- Using a substitution, that should be stated clearly, show that
$$\int \mathrm { f } ( x ) \mathrm { d } x = A \sinh ^ { - 1 } ( B x ) + c$$
where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
- Hence find, in exact form in terms of natural logarithms, the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\).