If $f(x)=\int\left[\tan ^2 x+\cot ^2 x+\frac{4\left(\sin ^3 x+\cos ^3 x\right)}{\sin ^2 2 x}\right] d x$ and $f\left(\frac{\pi}{4}\right)=0$, then $3\left[f\left(\frac{\pi}{6}\right)+2\right]=$

• Correct answer
  A

  $\frac{\pi}{2}$
• B

  $\frac{\pi}{4}$
• C

  0
• D

  $\frac{-\pi}{2}$

Hence, the correct answer is option: 'A'.