Isn't symmetry beautiful?

Calculus Level 2

\[\displaystyle\int_{-2014}^{2014}\begin{matrix}\color{white}{\mbox{~}}\\\color{white}{}\\\underbrace{\sin(\sin(\sin(\ldots\sin(\sin x)\ldots)))}\\\mbox{2014 times}\end{matrix} dx = \ ? \]

Give your answer to 3 decimal places.