I was trying to solve this integral and completely blanked on how to integrate vectors. I've only ever seen vector integrals like this $\int \vec r(t) dt$. This is straightforward as you just integrate each component wrt t and you have your new vector but for some reason I am struggling to integrate this:
$\int \sqrt{1+(d\vec r/dt)^2}dt$ where $\vec r = cos(2\pi t)i + sin(2\pi t)j+2\pi tk$.
Since $\dot{\vec{r}}=2\pi(-\sin(2\pi t)\vec{i}+\cos(2\pi t)\vec{j}+\vec{k})$, $\dot{\vec{r}}^2=8\pi^2$, making the integral $\sqrt{1+8\pi^2}\cdot t+C$.