- (a) Using the substitution \(u = \sqrt { 2 x + 1 }\), show that
$$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } \mathrm { e } ^ { \sqrt { 2 x + 1 } } \mathrm {~d} x$$
may be expressed in the form
$$\int _ { a } ^ { b } k u ^ { 2 } \mathrm { e } ^ { u } \mathrm {~d} u$$
where \(a\), \(b\) and \(k\) are constants to be found.
(b) Hence find, by algebraic integration, the exact value of
$$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } e ^ { \sqrt { 2 x + 1 } } d x$$
giving your answer in simplest form.