Answer
$$ 18.79$$
Work Step by Step
Given $$ \int_{2}^{4} \sqrt{x^{4}+1} d x, \quad N=8 $$
Since $\Delta x=\dfrac{b-a}{N}=\dfrac{1}{4}$ , 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}{12}\left[f(2)+4f(2.25)+2f(2.5)+4f(2.75)+2f(3) +4f(3.25)+2f(3.5) +4f(3.75)+f(4) \right]\\
&\approx 18.79
\end{align*}
