Integral of Infinite Sines

For $x_1 \in [0,\pi]$ the sequence defined by $ x_{n+1} = \sin x_n$ and $x_1$ converges to $0$: it is non negative and non increasing, hence converges. As $x \mapsto \sin x$ is continuous and $0$ is its only fix point, the limit is equal to zero.

Hence $ S_n$ converges pointwise to zero on $[0,\pi]$ and $$I=\int_0^{\pi}\lim_{n\rightarrow\infty}S_n\,dx = 0$$