If $f(x)=\begin{vmatrix}\cos x&1&0\\1&2\cos x&1\\0&1&2\cos x\end{vmatrix}$ then $\int\limits_0^{\large\frac{\pi}{2}}2f(x)dx$ is equal to

Question

$\begin{array}{1 1}(A)\;\large\frac{2}{3}&(B)\;\large\frac{-2}{3}\\(C)\;1&(D)\;-1\end{array}$