If $f(x)=\begin{vmatrix}1&a&a^2\\\sin(n-1) x&\sin nx&\sin(n+1)x\\\cos(n-1)x&\cos nx&\cos(n+1)x\end{vmatrix}$ then $\int_0^{\large\frac{\pi}{2}}f(x)dx$ is equal to

Question

$\begin{array}{1 1}(A)\;a-(1+a^2)&(B)\;1+a+a^2\\(C)\;-a+(1+a^2)&(D)\;-(1+a+a^2)\end{array}$