I am doing a homework and I am stuck on this problem. I need to find the lenght of a curve defined by
$$ x = \sqrt{2}t,\ y = e^t,\ z = e^{-t}. \quad 0 \leqslant t \leqslant 1 $$
I know that I need to differentiate each component, square them and finally add them up under a square root that I need to integrate. I come to this:
$$ \int_{1}^{0} \sqrt{2+e^{2t}+e^{-2t}} \, \mathrm{d}t, $$ but then I am stuck… I am pretty sure I did not make a mistake while doing differentiation, but I am not sure how to integrate this… If someone could help, I would be really grateful.
Hint:
$$\int_0^1\sqrt{2+e^{2t}+e^{-2t}}dt=\int_0^1\sqrt{\left(e^t+e^{-t}\right)^2}dt$$