8. (i) Find \(\int \tan ^ { 2 } x \mathrm {~d} x\).
(ii) Show that $$\int \tan x \mathrm {~d} x = \ln | \sec x | + c$$ where \(c\) is an arbitrary constant.
\includegraphics[max width=\textwidth, alt={}, center]{1e93a786-6105-4c69-a79a-a5f6e6c4aa0a-2_554_784_1484_507} The diagram 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 \(360 ^ { \circ }\) about the \(x\)-axis.
(iii) Show that the volume of the solid formed is \(\frac { 1 } { 18 } \pi ^ { 2 } ( 6 \sqrt { 3 } - \pi ) - \pi \ln 2\).