3 Fig. 8 shows part of the curve \(y = x \sin 3 x\). It crosses the \(x\)-axis at P . The point on the curve with \(x\)-coordinate \(\frac { 1 } { 6 } \pi\) is Q .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{74cc215f-bd55-489d-aa4b-0f67c2c8de52-2_420_780_549_655}
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\caption{Fig. 8}
\end{figure}
- Find the \(x\)-coordinate of P .
- Show that Q lies on the line \(y = x\).
- Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
- Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).
- Differentiate \(x \cos 2 x\) with respect to \(x\).
- Integrate \(x \cos 2 x\) with respect to \(x\).