9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-15_618_899_262_566}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
The curve \(C\) has parametric equations
$$x = \ln ( t + 2 ) , \quad y = \frac { 4 } { t ^ { 2 } } \quad t > 0$$
The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = \ln 3\) and \(x = \ln 5\)
- Show that the area of \(R\) is given by the integral
$$\int _ { 1 } ^ { 3 } \frac { 4 } { t ^ { 2 } ( t + 2 ) } \mathrm { d } t$$
- Hence find an exact value for the area of \(R\).
Write your answer in the form ( \(a + \ln b\) ), where \(a\) and \(b\) are rational numbers.
- Find a cartesian equation of the curve \(C\) in the form \(y = \mathrm { f } ( x )\).