7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ac7d862f-d10d-45ed-9077-ae4c7413cbf6-09_559_864_255_530}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The curve \(C\) has parametric equations
$$x = \ln ( t + 2 ) , \quad y = \frac { 1 } { ( t + 1 ) } , \quad t > - 1$$
The finite region \(R\) between the curve \(C\) and the \(x\)-axis, bounded by the lines with equations \(x = \ln 2\) and \(x = \ln 4\), is shown shaded in Figure 3.
- Show that the area of \(R\) is given by the integral
$$\int _ { 0 } ^ { 2 } \frac { 1 } { ( t + 1 ) ( t + 2 ) } \mathrm { d } t$$
- Hence find an exact value for this area.
- Find a cartesian equation of the curve \(C\), in the form \(y = \mathrm { f } ( x )\).
- State the domain of values for \(x\) for this curve.
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