Problem

(a) Using the substitution $u=\sqrt{2x+1}$, show that

$$\int_4^{12}\sqrt{8x+4}\,e^{\sqrt{2x+1}}\,dx$$

may be expressed in the form

$$\int_a^b ku^2e^u\,du$$

where $a$, $b$ and $k$ are constants to be found.

(b) Hence find, by algebraic integration, the exact value of

$$\int_4^{12}\sqrt{8x+4}\,e^{\sqrt{2x+1}}\,dx$$

giving your answer in simplest form.