Question

Evaluate the integral: \[ \int_0^{\frac{\pi}{4}} \frac{\cos^2 x}{\cos^2 x + 4 \sin^2 x} \, dx \]

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

Hint:

For integrals involving trigonometric functions, use substitution or trigonometric identities to simplify the expression before integrating.

Answer 1

The Correct Option is C

We are given the integral \( \int_0^{\frac{\pi}{4}} \frac{\cos^2 x}{\cos^2 x + 4 \sin^2 x} \, dx \). Step 1: Rewrite the integral using trigonometric identities. The denominator can be rewritten as a sum involving the tangent function. Step 2: Apply substitution to simplify the expression and evaluate the integral. After solving, the result is: \[ -\frac{\pi}{12} + \frac{2}{3} \tan^{-1} 2 \] % Final Answer The value of the integral is \( -\frac{\pi}{12} + \frac{2}{3} \tan^{-1} 2 \).