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\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-06_544_1232_251_260} The diagram shows the curve with parametric equations \(x = \ln \left( t ^ { 2 } - 4 \right) , \quad y = \frac { 4 } { t ^ { 2 } } , \quad\) where \(t > 2\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = \ln 5\) and \(x = \ln 12\).

1. Show that the area of \(R\) is given by
   \(\int _ { a } ^ { b } \frac { 8 } { t \left( t ^ { 2 } - 4 \right) } \mathrm { d } t\),
   where \(a\) and \(b\) are constants to be determined.
2. In this question you must show detailed reasoning. Hence find the exact area of \(R\), giving your answer in the form \(\ln k\), where \(k\) is a constant to be determined.
3. Find a cartesian equation of the curve in the form \(y = \mathrm { f } ( x )\).