4. (a) Express \(\frac { 25 } { x ^ { 2 } ( 2 x + 1 ) }\) in partial fractions.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e14881c1-5ba5-4868-92ee-8bc58d4884dc-06_623_849_408_561}
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\caption{Figure 2}
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Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = \frac { 5 } { x \sqrt { } ( 2 x + 1 ) } , x > 0\)
The finite region \(R\) is bounded by the curve \(C\), the \(x\)-axis, the line with equation \(x = 1\) and the line with equation \(x = 4\)
This region is shown shaded in Figure 2
The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
(b) Use calculus to find the exact volume of the solid of revolution generated, giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are constants.