11. (a) Given
$$\frac { 9 } { t ^ { 2 } ( t - 3 ) } \equiv \frac { A } { t } + \frac { B } { t ^ { 2 } } + \frac { C } { ( t - 3 ) }$$
find the value of the constants \(A , B\) and \(C\).
(b)
$$I = \int _ { 4 } ^ { 12 } \frac { 9 } { t ^ { 2 } ( t - 3 ) } \mathrm { d } t$$
Find the exact value of \(I\), giving your answer in the form \(\ln ( a ) - b\), where \(a\) and \(b\) are positive constants.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-34_535_880_959_525}
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\caption{Figure 3}
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Figure 3 shows a sketch of part of the curve \(C\) with parametric equations
$$x = 2 \ln ( t - 3 ) , \quad y = \frac { 6 } { t } \quad t > 3$$
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = 2 \ln 9\)
The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
(c) Show that the exact volume of the solid generated is
$$k \times I$$
where \(k\) is a constant to be found.