# Arc length problem with a thorny integration

#### aucuneidee

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

So, the question gives a particle travelling over a path $\gamma$, and I need the arc length.

2. Relevant equations

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

We want the arc length over $1 \le t \le 4$.

3. The attempt at a solution

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

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

But that's where the fun ends. I've tried a bunch of trig substitutions (e.g. $t=\tan u$), 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!

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#### HallsofIvy

Didn't you notice that $t^2+ 1+ 1/t^2= t^2+ 2+ 1/t^2- 1= (t+ 1/t)^2- 1$?

#### aucuneidee

Didn't you notice that $t^2+ 1+ 1/t^2= t^2+ 2+ 1/t^2- 1= (t+ 1/t)^2- 1$?
I had, but didn't realise it would help. I'll play around and see what I can come up with. :)

Thankyou for the quick response!

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