Which of the following integrals represents the length of the parametric curve $x=t+\cos(t),\;\;y=t-\sin(t),\;\;0 \le t \le 2\pi$ ?

A. $\displaystyle \int_{0}^{\,2\pi} {\sqrt{4-2\sin(t)-2\cos(t)}}\,dt$
B. $\displaystyle \int_{0}^{\,2\pi} {\sqrt{2-\sin(t)-\cos(t)}}\,dt$
C. $\displaystyle \int_{0}^{\,2\pi} {\sqrt{3-\sin(t)-\cos(t)}}\,dt$
D. $\displaystyle \int_{0}^{\,2\pi} {\sqrt{4-\sin(t)-\cos(t)}}\,dt$
E. $\displaystyle \int_{0}^{\,2\pi} {\sqrt{3-2\sin(t)-2\cos(t)}}\,dt$
F. $\displaystyle \int_{0}^{\,2\pi} {\sqrt{2-2\sin(t)-2\cos(t)}}\,dt$