Standard integral of 1/√(x²-a²)

1 Find the value of \(\int _ { 6 } ^ { 7 } \frac { 1 } { \sqrt { ( x - 5 ) ^ { 2 } - 1 } } \mathrm {~d} x\), giving your answer in the form \(\ln ( a + \sqrt { b } )\), where \(a\) and \(b\) are integers to be determined.

12. (a) Evaluate \(\int _ { 3 } ^ { 4 } \frac { 1 } { \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x\), giving your answer correct to three decimal places.
(b) Given that \(\int _ { 1 } ^ { 2 } \frac { k } { 9 - x ^ { 2 } } \mathrm {~d} x = \ln \frac { 25 } { 4 }\), find the value of \(k\).
(c) Show that \(\int \frac { ( \cosh x - \sinh x ) ^ { 3 } } { \cosh ^ { 2 } x + \sinh ^ { 2 } x - \sinh 2 x } \mathrm {~d} x\) can be expressed as \(- \mathrm { e } ^ { - x } + c\), where \(c\) is a constant.