$\int_0^{\pi / 4} \frac{\cos ^2 x}{\cos ^2 x+4 \sin ^2 x} d x=$

• A

  $\frac{\pi}{2}-\frac{1}{3} \operatorname{Tan}^{-1} 2$
• B

  $-\frac{\pi}{4}-\frac{4}{3} \operatorname{Tan}^{-1} 2$
• C

  $\frac{\pi}{6}+\frac{2}{3} \operatorname{Tan}^{-1} 2$
• Correct answer
  D

  $-\frac{\pi}{12}+\frac{2}{3} \operatorname{Tan}^{-1} 2$

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