5 The curve \(C\) has polar equation \(r = \ln ( 1 + \pi - \theta )\), for \(0 \leqslant \theta \leqslant \pi\).
- Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
- Using the substitution \(u = 1 + \pi - \theta\), or otherwise, show that the area of the region enclosed by \(C\) and the initial line is
$$\frac { 1 } { 2 } ( 1 + \pi ) \ln ( 1 + \pi ) ( \ln ( 1 + \pi ) - 2 ) + \pi$$
- Show that, at the point of \(C\) furthest from the initial line,
$$( 1 + \pi - \theta ) \ln ( 1 + \pi - \theta ) - \tan \theta = 0$$
and verify that this equation has a root between 1.2 and 1.3.