Chapter 8.8, Problem 51E

Simpson’s Rule Apply Simpson’s Rule to the following integrals. It is easiest to obtain the Simpson’s Rule approximations from the Trapezoid Rule approximations, as in Example 7. Make a table similar to Table 7.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

41. ${\displaystyle {\int }_0^{\pi }{e}^{-t}\sin tdt=\frac{1}{2}(e^{-\pi }+1)}$