IMC 1998 · Problem 3

Let $f(x) = 2x(1-x)$, $x \in \mathbb{R}$. Define

$$f_n = \overbrace{f \circ \dots \circ f}^{n}.$$

a) Find $\lim\limits_{n \to \infty} \int_0^1 f_n(x)\,dx$.

b) Compute $\int_0^1 f_n(x)\,dx$ for $n = 1, 2, \dots$.