Answer
$$2.5450$$
Work Step by Step
Given $$ \int_{1}^{4} \ln x d x, \quad N=8 $$
Since $\Delta x=\dfrac{b-a}{N}=\dfrac{3}{8}$ , 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) +f(x_{6}) \right] \\
&=\dfrac{1}{8}\left[f(1)+4f(11/8 )+2f(14/8)+4f(17/8)+2f(20/8) +4f(23/8)+2f(26/8)+4f(29/8)+f(4) \right]\\
&\approx 2.5416
\end{align*}
