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**1. The problem statement, all variables and given/known data**

So, the question gives a particle travelling over a path [itex]\gamma[/itex], and I need the arc length.

**2. Relevant equations**

The path is [itex]\gamma(t) : [1,4] \to ℝ^3, t \mapsto (t^2/2, t, ln(2t))[/itex].

We want the arc length over [itex]1 \le t \le 4[/itex].

**3. The attempt at a solution**

First, the speed differential: [itex] ds = \left\| \gamma'(t) \right\| dt = \sqrt{t^2 + 1 + 1 /t^2} dt [/itex]

Now, the arc length. [itex]\ell = \int_\gamma ds = \int_1^4 \sqrt{t^2 + 1 + 1 /t^2}dt[/itex].

But that's where the fun ends. I've tried a bunch of trig substitutions (e.g. [itex]t=\tan u[/itex]), to no avail.

I also tried Wolfram online integrator, which returned a mess of symbols -- this problem should have a (reasonably) simple analytic solution.

Any ideas, anyone? I'd really appreciate any help!