MathNet 2004 — Part 03 Question 9

Let $P(x) = x^{3} - \frac{3}{2} x^{2} + x + \frac{1}{4}$. Let $P^{[1]}(x) = P(x)$, and for $n \geq 1$, let $P^{[n+1]}(x) = P^{[n]}(P(x))$. Evaluate $\int_{0}^{1} P^{[2004]}(x) \, d x$.