Express \(3 \cos 2 x - \sqrt { 3 } \sin 2 x\) in the form \(R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact values of \(R\) and \(\alpha\).
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Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } \frac { 3 } { ( 3 \cos 2 x - \sqrt { 3 } \sin 2 x ) ^ { 2 } } \mathrm {~d} x\), simplifying your answer.