I am trying to do ...

$\displaystyle \int_0^{2\pi}\sqrt{1-cos(t)}dt$

Here is what I did. Can you either comment on my work or post a better solution?

Assume the domain of all functions is$\displaystyle (0,\pi)$.Let$\displaystyle u=1-cos(t)$

$\displaystyle \displaystyle\int\sqrt{1-cos(x)}dx=\int\frac{sin(x)}{\sqrt{1+cos(x)}}dx=\in t\frac{1}{\sqrt{2-u}}du=-2\sqrt{1+cos(x)}$

Therefore,$\displaystyle \int_0^\pi\sqrt{1-cos(x)}dx=2\sqrt{2}$

Since the integrand is symmetric about$\displaystyle y=\pi$,$\displaystyle \int_0^{2\pi}\sqrt{1-cos(t)}dt=2*2\sqrt{2}$