| Exam Board | SPS |
|---|
| Module | SPS SM Pure (SPS SM Pure) |
|---|
| Year | 2025 |
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| Session | February |
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| Topic | Integration by Parts |
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6.
The region bounded by the curve
$$y = ( 2 x - 8 ) \ln x$$
and the \(x\)-axis is shaded in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-12_871_913_422_575}
Show that the exact area is given by
$$32 \ln 2 - \frac { 33 } { 2 }$$
Fully justify your answer.