Prove, from definitions involving exponential functions, that
$$\cosh 2 u = 2 \sinh ^ { 2 } u + 1$$
Prove that, if \(y \geqslant 0\) and \(\cosh y = u\), then \(y = \ln \left( u + \sqrt { } \left( u ^ { 2 } - 1 \right) \right)\).
Using the substitution \(2 x = \cosh u\), show that
$$\int \sqrt { 4 x ^ { 2 } - 1 } \mathrm {~d} x = a x \sqrt { 4 x ^ { 2 } - 1 } - b \operatorname { arcosh } 2 x + c$$
where \(a\) and \(b\) are constants to be determined and \(c\) is an arbitrary constant.
Find \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \sqrt { 4 x ^ { 2 } - 1 } \mathrm {~d} x\), expressing your answer in an exact form involving logarithms.