Find the exact value of \(\int _ { 1 } ^ { 2 } \frac { 2 } { ( 4 x - 1 ) ^ { 2 } } \mathrm {~d} x\).
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The diagram shows part of the curve \(y = \frac { 1 } { x }\). The point \(P\) has coordinates \(\left( a , \frac { 1 } { a } \right)\) and the point \(Q\) has coordinates \(\left( 2 a , \frac { 1 } { 2 a } \right)\), where \(a\) is a positive constant. The point \(R\) is such that \(P R\) is parallel to the \(x\)-axis and \(Q R\) is parallel to the \(y\)-axis. The region shaded in the diagram is bounded by the curve and by the lines \(P R\) and \(Q R\). Show that the area of this shaded region is \(\ln \left( \frac { 1 } { 2 } \mathrm { e } \right)\).