Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\ln{\left(x \right)}$ on $LaTeX: \displaystyle (1,2)$ with $LaTeX: \displaystyle n=34$ .

$LaTeX: \displaystyle \Delta x = \frac{ 2 - 1 }{ 34 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 1 + i \frac{1}{34}$ Using the 1,4,2,...,2,4,1 pattern the sum can be written as $LaTeX: \displaystyle x_i$ can be written split into the even and odd terms. $LaTeX: \displaystyle x_k = 1 + (2k-1)\cdot \frac{1}{34}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =17$ and $LaTeX: \displaystyle x_j = 1 + (2j)\cdot \frac{1}{34}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =16$ . $LaTeX: \displaystyle f(1) +f(2)+4\sum_{k=1}^{17}f\left(\frac{k}{17} + \frac{33}{34}\right) + 2\sum_{j=1}^{16}f\left(\frac{j}{17} + 1\right)$ . The value is $LaTeX: \displaystyle 1.222$