Approximate the integral using Simpson's rule $S_{10}$ and compare your answer to that produced by a calculating utility with a numerical integration capability.
Express your final approximation answers to six decimal places.

$\displaystyle \int_{0}^{\pi } \frac{x}{1+\sin\!\left(x\right)} \;dx=$ (from CAS).

$S_{10}=$ , $E_{S}=$