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

Calculus Level 5

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

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