Question

If \[ \int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 16 \sin 2x} \, dx = k \log 3, \text{ then } k = \]

• A. \( \frac{1}{30} \)
• B. \( \frac{1}{20} \)
• C. \( \frac{1}{10} \)
• D. \( \frac{1}{40} \)

Hint:

When solving integrals with trigonometric functions, use trigonometric identities to simplify the expressions before integration.

Answer 1

The Correct Option is B

Step 1: Simplify the integral.
We are given the integral: \[ \int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 16 \sin 2x} \, dx \] We know that \( \sin 2x = 2 \sin x \cos x \), so the denominator becomes: \[ 9 + 16 \sin 2x = 9 + 32 \sin x \cos x \] Thus, the integral simplifies to: \[ \int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 32 \sin x \cos x} \, dx \]
Step 2: Solve the integral.
The integral is simplified and computed, giving the result \( k = \frac{1}{20} \), corresponding to option (B).