2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ef3ae4a-a06d-48c1-8b79-7d7c3f95d120-03_623_1176_196_374}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve with equation \(y = x \ln x , x \geqslant 1\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line \(x = 4\).
The table shows corresponding values of \(x\) and \(y\) for \(y = x \ln x\).
| \(x\) | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
| \(y\) | 0 | 0.608 | | | 3.296 | 4.385 | 5.545 |
- Complete the table with the values of \(y\) corresponding to \(x = 2\) and \(x = 2.5\), giving your answers to 3 decimal places.
- Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
- Use integration by parts to find \(\int x \ln x \mathrm {~d} x\).
- Hence find the exact area of \(R\), giving your answer in the form \(\frac { 1 } { 4 } ( a \ln 2 + b )\), where \(a\) and \(b\) are integers.