Given that \(\frac { 3 } { 9 - x ^ { 2 } }\) can be expressed in the form \(k \left( \frac { 1 } { 3 + x } + \frac { 1 } { 3 - x } \right)\), find the value of the rational number \(k\).
Show that \(\int _ { 1 } ^ { 2 } \frac { 3 } { 9 - x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.