(i) Show that \(5+4\tan^2\left(\dfrac{x}{3}\right)=4\operatorname{sec}^2\left(\dfrac{x}{3}\right)+1\).

(ii) Given that \(\dfrac{d}{dx}\left(\tan\left(\dfrac{x}{3}\right)\right)=\dfrac13\operatorname{sec}^2\left(\dfrac{x}{3}\right)\), find \(\displaystyle\int \operatorname{sec}^2\left(\dfrac{x}{3}\right)\,dx\).

(iii) The diagram shows part of the curve \(y=5+4\tan^2\left(\dfrac{x}{3}\right)\). Using the results from parts (i) and (ii), find the exact area of the shaded region enclosed by the curve, the \(x\)-axis and the lines \(x=\dfrac{\pi}{2}\) and \(x=\pi\).