Standard +0.3 This is a straightforward integration by parts question requiring students to find where the curve crosses the x-axis (x=1 and x=4), set up the integral with correct limits accounting for the curve being below the axis, apply integration by parts to ∫(2x-8)ln(x)dx, and evaluate. While it requires multiple steps and careful attention to signs, it follows a standard template with no novel insights needed—slightly easier than average due to the clear setup and routine application of a single technique.
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.
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.\\
\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q6 [4]}}