integral of derivated function mix $$\int_0^{2\pi}(t-\sin t){\sqrt{1-\cos t}} dt$$
I can notice that I have something of the form 
$$\int{f(x){\sqrt {f'(x)}}dx}$$ but I don't know anything that could simplify it

HINT:
Note that $\sqrt{1-\cos(x)}=\sqrt{2}|\sin(x/2)|$
The integral of $x\sin(x/2)$ can be evaluated using integration by parts, while the integral of $\sin(x)\sin(x/2)$ is facilitated using the addition angle formula.
