Partial fractions parametric area

A question is this type if and only if converting a parametric area integral into parameter form produces a rational function that must be decomposed using partial fractions before integration.

2 questions · Standard +0.3

1.03g Parametric equations: of curves and conversion to cartesian1.08h Integration by substitution
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Edexcel C34 2015 January Q9
12 marks Standard +0.3
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 \(x = \ln 5\)
  1. Show that the area of \(R\) is given by the integral $$\int _ { 1 } ^ { 3 } \frac { 4 } { t ^ { 2 } ( t + 2 ) } \mathrm { d } t$$
  2. 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.
  3. Find a cartesian equation of the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
Edexcel C4 2008 January Q7
15 marks Standard +0.3
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.
  1. Show that the area of \(R\) is given by the integral $$\int _ { 0 } ^ { 2 } \frac { 1 } { ( t + 1 ) ( t + 2 ) } \mathrm { d } t$$
  2. Hence find an exact value for this area.
  3. Find a cartesian equation of the curve \(C\), in the form \(y = \mathrm { f } ( x )\).
  4. State the domain of values for \(x\) for this curve. \(\_\_\_\_\)}