Question

Evaluate the integral \( \int_0^{\pi} \frac{x \sin x}{4 \cos^2 x + 3 \sin^2 x} dx \):

• A. \( \frac{\pi^2}{6\sqrt{3}} \)
• B. \( \frac{\pi}{3\sqrt{3}} \)
• C. \( \frac{\pi^2}{3\sqrt{3}} \)
• D. \( \sqrt{3}\pi^2 \)

Hint:

For integrals involving trigonometric functions, often simplifying using known trigonometric identities or standard results helps in evaluating the integral more easily.

Answer 1

The Correct Option is A

We are given the integral \[ I = \int_0^{\pi} \frac{x \sin x}{4 \cos^2 x + 3 \sin^2 x} dx. \] 

Step 1: Simplify the denominator using a standard trigonometric identity: \[ 4 \cos^2 x + 3 \sin^2 x = 4 - \sin^2 x. \] 

Step 2: The integral can be simplified further, but we can use known results for trigonometric integrals to directly obtain: \[ I = \frac{\pi^2}{6\sqrt{3}}. \] Thus, the correct answer is \( \frac{\pi^2}{6\sqrt{3}} \).