10. 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 $\int_{0}^{1} P^{[2004]}(x) d x$.
Solution: $1 / 2$
By Note that $P(1-x)=1-P(x)$. It follows easily by induction that $P^{[k]}(1-x)=$ $1-P^{[k]}(x)$ for all positive integers $k$. Hence
$$
\begin{aligned}
\int_{0}^{1} P^{[2004]}(x) d x & =\int_{0}^{1} 1-P^{[2004]}(1-x) d x \\
& =1-\int_{0}^{1} P^{[2004]}(1-x) d x \\
& =1-\int_{0}^{1} P^{[2004]}(u) d u \quad(u=1-x) .
\end{aligned}
$$

Hence $\int_{0}^{1} P^{[2004]}(x) d x=1 / 2$.