Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

$$I_n = \int_1^6 x^n(3x-2)^2\,dx \qquad n\ge 0$$

(a) Show that, for $n \ge 1$

$$(3+6n)I_n = 4nI_{n-1} + 8\cdot 6^{\,n-2}$$

(b) Use the reduction formula in part (a) to determine the exact value of

$$\int_1^6 x^3(3x-2)^2\,dx$$