Arc Length of $\ln\left(x\right)$

Calculus Level 5

Let $f(n)$ denote the arc length of the curve $y=\ln\left(x\right)$ in the interval $x\in\left [1,n\right ]$ . Find $\lim_{n\to\infty}\big(n-f(n)\big).$ If your answer is of the form $\ln\left(\sqrt{a}-b\right)+\sqrt{c}-d$ , where $a$ , $b$ , $c$ , and $d$ are non-negative integers and $a$ and $c$ are not perfect squares, find $a+b+c+d$ .

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Arc Length of ln⁡(x)\ln\left(x\right)ln(x)

Arc Length of $\ln\left(x\right)$