Question

Evaluate \( \int_{0}^{\pi/4} \frac{\cos^2 x}{\cos^2 x + 4 \sin^2 x} \, dx \):

• A. \( \frac{\pi}{4} + \frac{2}{3} \tan^{-1} 2 \)
• B. \( -\frac{\pi}{3} + \frac{2}{3} \tan^{-1} 3 \)
• C. \( -\frac{\pi}{12} + \frac{2}{3} \tan^{-1} 2 \)
• D. \( \frac{\pi}{6} - \frac{2}{3} \tan^{-1} 4 \)

Hint:

Substituting \( \tan x = t \) in integrals involving trigonometric functions can help simplify the expression and convert it into a rational function.