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\includegraphics[max width=\textwidth, alt={}, center]{05bec6d6-b526-4b6f-86f3-39aa38cbf5c6-7_540_734_260_667}
The diagram shows part of the curve \(y = \ln ( x - 4 )\).
- Use integration by parts to show that \(\int \ln ( x - 4 ) \mathrm { d } x = ( x - 4 ) \ln | x - 4 | - x + c\).
- State the equation of the vertical asymptote to the curve \(y = \ln ( x - 4 )\).
- Find the total area enclosed by the curve \(y = \ln ( x - 4 )\), the \(x\)-axis and the lines \(x = 4.5\) and \(x = 7\). Give your answer in the form \(a \ln 3 + b \ln 2 + c\) where \(a , b\) and \(c\) are constants to be found.