2 The curve \(C\) has polar equation \(r^{2}=\ln (1+\theta)\), for \(0 \leqslant \theta \leqslant 2 \pi\).
(i) Sketch \(C\).

(ii) Using the substitution \(u=1+\theta\), or otherwise, find the area of the region bounded by \(C\) and the initial line, leaving your answer in an exact form.