Let \(I = \int_{\frac{1}{4}}^{\frac{3}{4}} \sqrt{\left( \frac{x}{1-x} \right)} \, dx\).

(i) Using the substitution \(x = \cos^2 \theta\), show that \(I = \int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi} 2 \cos^2 \theta \, d\theta\).

(ii) Hence find the exact value of \(I\).