Normal/tangent then area with parts

10 \includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-18_337_529_260_808} The diagram shows the curve \(y = ( \ln x ) ^ { 2 }\). The \(x\)-coordinate of the point \(P\) is equal to e, and the normal to the curve at \(P\) meets the \(x\)-axis at \(Q\).

1. Find the \(x\)-coordinate of \(Q\).
2. Show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\), where \(c\) is a constant.
3. Using integration by parts, or otherwise, find the exact value of the area of the shaded region between the curve, the \(x\)-axis and the normal \(P Q\).

8 \includegraphics[max width=\textwidth, alt={}, center]{dde12c57-5129-43ae-b385-9a8f21f51e49-3_566_787_255_680} The diagram shows the curve \(y = x \sin x\), for \(0 \leqslant x \leqslant \pi\). The point \(Q \left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi \right)\) lies on the curve.

1. Show that the normal to the curve at \(Q\) passes through the point \(( \pi , 0 )\).
2. Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin x - x \cos x )\).
3. Hence evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).

8 \includegraphics[max width=\textwidth, alt={}, center]{2aceb797-097c-499b-99b6-cce9f287cb51-3_566_787_255_680} The diagram shows the curve \(y = x \sin x\), for \(0 \leqslant x \leqslant \pi\). The point \(Q \left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi \right)\) lies on the curve.

1. Show that the normal to the curve at \(Q\) passes through the point \(( \pi , 0 )\).
2. Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin x - x \cos x )\).
3. Hence evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).

8 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] \end{figure}

1. Find the \(x\)-coordinate of P .
2. Show that Q lies on the line \(y = x\).
3. Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).

2 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] \end{figure}

1. Find the \(x\)-coordinate of P .
2. Show that Q lies on the line \(y = x\).
3. Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).

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] \end{figure}

1. Find the \(x\)-coordinate of P .
2. Show that Q lies on the line \(y = x\).
3. Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).
5. Differentiate \(x \cos 2 x\) with respect to \(x\).
6. Integrate \(x \cos 2 x\) with respect to \(x\).

8 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] \end{figure}

1. Find the \(x\)-coordinate of P .
2. Show that Q lies on the line \(y = x\).
3. Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).

1. a. Find \(\int \ln x \mathrm {~d} x\)

\begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = \ln x , \quad x > 0$$ The point P lies on \(C\) and has coordinate \(( e , 1 )\).
The line 1 is a normal to \(C\) at \(P\). The line \(l\) cuts the \(x\)-axis at the point \(Q\).
b. Find the exact value of the \(x\)-coordinate of \(Q\). The finite region \(\mathbf { R }\), shown shaded in figure 3, is bounded by the curve, the line \(l\) and the \(x\)-axis.
c. Find the exact area of \(\mathbf { R }\).

7 A curve has equation \(y = 4 x \cos 2 x\).

1. Find an exact equation of the tangent to the curve at the point on the curve where $$x = \frac { \pi } { 4 }$$
2. The region shaded on the diagram below is bounded by the curve \(y = 4 x \cos 2 x\) and the \(x\)-axis from \(x = 0\) to \(x = \frac { \pi } { 4 }\). \includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-8_487_878_740_591} By using integration by parts, find the exact value of the area of the shaded region.
   (5 marks)
   \includegraphics[max width=\textwidth, alt={}]{b8614dd6-2197-40c3-a673-5bef3e3653a5-8_1275_1717_1432_150}