HMMT 2004 K10

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 $\int_{0}^{1} P^{[2004]}(x) d x$.