5. A curve has parametric equations
$$x = \frac { t } { 2 - t } , \quad y = \frac { 1 } { 1 + t } , \quad - 1 < t < 2$$
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { 2 } \left( \frac { 2 - t } { 1 + t } \right) ^ { 2 }\).
- Find an equation for the normal to the curve at the point where \(t = 1\).
- Show that the cartesian equation of the curve can be written in the form
$$y = \frac { 1 + x } { 1 + 3 x }$$
- continued
- (a) Find \(\int \tan ^ { 2 } x d x\).
- Show that
$$\int \tan x \mathrm {~d} x = \ln | \sec x | + c$$
where \(c\) is an arbitrary constant.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fe01157a-7617-43d3-900c-8d043bcbe784-10_566_789_648_504}
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\caption{Figure 1}
\end{figure}
Figure 1 shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \tan x\).
The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 3 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis. - Show that the volume of the solid formed is \(\frac { 1 } { 18 } \pi ^ { 2 } ( 6 \sqrt { 3 } - \pi ) - \pi \ln 2\).