Arc Length parametrized by $r(t) = (\cos(e^t),\sin(e^t),e^t)$

Question

Using the vector valued function given by: $\mathbf r(t)=[\cos(e^t), \sin(e^t), e^t]$, for $0 \le t \le t$, compute the arc length.

I have the derivatives of each component of the vector and I know that the arc length is computed by taking the integral of the square root of the distance formula. I just don't know how to how to simplify what is under the square root in order to take the integral. Please help!

Answer 1

$\dot{r}(t)=(-\sin(e^t)\cdot e^t,\cos(e^t)\cdot e^t,e^t)$,

$\|\dot{r}(t)\|^2=\sin^2(e^t)\cdot e^{2t}+\cos^2(e^t)\cdot e^{2t}+e^{2t}=2e^{2t}$,

$\|\dot{r}(t)\|=\sqrt{2}\cdot e^t$.