Answer
$$0.6087$$
Work Step by Step
Given $$\int_{5}^{9} \cos \left(x^{2}\right) d x$$
Since $\Delta x=\dfrac{b-a}{N}=\dfrac{4}{8}=0.5$ , then by using Simpson’s rule
\begin{align*}
S_{n}&=\dfrac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)..+4f(x_{n-1})+f(x_n)\right]\\
S_{8}&=\dfrac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+4f(x_5) +2f(x_{6}) +4f(x_{7})+f(x_{8})\right] \\
&=\dfrac{1}{6}\left[f(5)+4f(5.5)+2f(6)+4f(6.5)+2f(7) +4f(7.5)+2f(8)+4f(8.5)+f(9) \right]\\
&\approx 0.6087
\end{align*}
