Repeated factor with x² in denominator

4. (a) Express \(\frac { 25 } { x ^ { 2 } ( 2 x + 1 ) }\) in partial fractions. \begin{figure}[h] \end{figure} 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.

1 Express \(\frac { x + 1 } { x ^ { 2 } ( 2 x - 1 ) }\) in partial fractions.

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1. Express \(\frac { 4 x + 3 } { ( x - 1 ) ^ { 2 } }\) in the form \(\frac { A } { x - 1 } + \frac { B } { ( x - 1 ) ^ { 2 } }\) 7
2. Show that $$\int _ { 3 } ^ { 4 } \frac { 4 x + 3 } { ( x - 1 ) ^ { 2 } } \mathrm {~d} x = p + \ln q$$ where \(p\) and \(q\) are rational numbers.