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If $f(x) = \begin{vmatrix} 2 \cos ^2 x & \sin 2x & – \sin x \\ \sin 2x & 2 \sin ^2 x & \cos x \\ \sin x & – \cos x & 0 \end{vmatrix},$  then $\int_0^{\frac{\pi}{2}} [ f(x) + f’(x)] dx$  is

  1. $\pi$
  2. $\frac{\pi}{2}$
  3. $0$
  4. $1$
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