Find the length of curve using the integral

Question

Find the length of curve $$x(t)=1+2\cos(t)+\cos(2t)$$ $$y(t)=2\sin(t)+\sin(2t)$$

$$0<t<2π$$

I actually couldn't get the integral after writing the equation..

edit:sorry it must be 2π, not π

Answer 1

Let $z:=x+iy=1+2e^{it}+e^{2it}$ so$$\begin{align}ds&=\sqrt{\dot{z}\dot{z}^\ast}dt\\&=\sqrt{(2ie^{it}+2ie^{2it})(-2ie^{-it}-2ie^{-2it})}dt\\&=2\sqrt{(1+e^{it})(1+e^{-it})}dt\\&=2\sqrt{2(1+\cos t)}dt\\&=|4\cos\tfrac{t}{2}|dt.\end{align}$$So the arclength is$$\begin{align}\int_0^{2\pi}4|\cos\tfrac{t}{2}dt|&=\int_0^{\pi}8\cos\tfrac{t}{2}dt\\&=\left[16\sin\tfrac{t}{2}\right]_0^\pi\\&=16.\end{align}$$