8 The constant \(a\), where \(a > 1\), is such that \(\int _ { 1 } ^ { a } \left( x + \frac { 1 } { x } \right) \mathrm { d } x = 6\).
- Find an equation satisfied by \(a\), and show that it can be written in the form
$$a = \sqrt { } ( 13 - 2 \ln a )$$
- Verify, by calculation, that the equation \(a = \sqrt { } ( 13 - 2 \ln a )\) has a root between 3 and 3.5.
- Use the iterative formula
$$a _ { n + 1 } = \sqrt { } \left( 13 - 2 \ln a _ { n } \right)$$
with \(a _ { 1 } = 3.2\), to calculate the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.