Answer
$$ 0.947$$
Work Step by Step
Given$$ \int_{0}^{1} \cos \left(x^{2}\right) d x, \quad N=6$$
Since $\Delta x=\dfrac{b-a}{N}=\dfrac{1}{6}$ , 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_{6}&=\dfrac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+4f(x_5) +f(x_{6}) \right] \\
&=\dfrac{1}{18}\left[f(0)+4f(1/6)+2f(2/6)+4f(3/6)+2f(4/6) +4f(5/6)+f(1) \right]\\
&\approx 0.947
\end{align*}
