> 
> *Determine the arc length of the parametric curve given by the following parametric equation:
>  $φ(t)= (\sqrt{t}, t+1, t)$ $t\in[10,20]$*
> 
> 
> 

In order to do this I simply tried it to solve it by the formula of arc lenght. 

Given the formula, $
L=\int\_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 +\left(\frac{dy}{dt}\right)^2 +\left(\frac{dz}{dt}\right)^2 } \;dt$

I get that $L=\int\_{10}^{20} \sqrt {2+\frac{1}{4t}}; dt=\int\_{10}^{20}\sqrt {\frac{8t+1}{4t}} \; dt$ 

Then, I tried to get prettier the integral function by multiplying in the argument of the square root by $4t/4t$ (I eliminated square root in the denominator) getting this not very satisfying result:

$L=\int\_{10}^{20}\sqrt {\frac{(4t)^2+t}{2t}} \; dt$ 

I really don't know how to solve that integral, so I wonder if someone comes out with some idea about how to do it. 

In the other hand, I tought that maybe there is a nicer parametrization of that particular parametrization, e.g. a reparametrization, where we could solve the integral more easily, but in that case I'm not sure about how to find it. Any ideas?