Arc length of parametric curves

Problem 8.92 · medium

1. \[ \left[\begin{matrix}\frac{d}{d t} e^{t} \cos{\left(t \right)}\\\frac{d}{d t} e^{t} \sin{\left(t \right)}\end{matrix}\right] = \left[\begin{matrix}- e^{t} \sin{\left(t \right)} + e^{t} \cos{\left(t \right)}\\e^{t} \sin{\left(t \right)} + e^{t} \cos{\left(t \right)}\end{matrix}\right] \]
   Velocity components.✓ Proved
2. \[ \left(- e^{t} \sin{\left(t \right)} + e^{t} \cos{\left(t \right)}\right)^{2} + \left(e^{t} \sin{\left(t \right)} + e^{t} \cos{\left(t \right)}\right)^{2} = 2 e^{2 t} \]
   (dx/dt)² + (dy/dt)², simplified.✓ Proved
3. \[ \int\limits_{0}^{1} \sqrt{2} e^{t}\, dt = \sqrt{2} \left(-1 + e\right) \]
   Integrate the speed.✓ Proved