Find the arc length of the curve $r(t) = \left<\frac{2t}{1+t^2}, \frac{1-t^2}{1+t^2}\right>$ between $t = 0$, and $t = 2$.
I know that first step is to differentiate and then take the integral of the magnitude of differential from $0$ to $2.$ But When I take the differential of $r(t)$ I'm getting quite a long expression. And the magnitude of that expression is even longer, I'm sure there's a better (shorter) way of doing this. Any help is appreciated.
Thanks in advance.
One method is to put $t = \tan \theta$. Then it reduces $$r(\theta) =\langle \sin(2\theta), \cos(2\theta)\rangle$$