Question

The value of \[ \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{1}{1 + \sqrt{\tan 2x}} \, dx \] is:

• A. \( \frac{\pi}{12} \)
• B. \( \frac{\pi}{6} \)
• C. \( \frac{\pi}{24} \)
• D. \( \frac{\pi}{3} \)

Hint:

For integrals involving trigonometric functions, using substitution and trigonometric identities can help simplify the problem and make the integration process easier.

Answer 1

The Correct Option is A

Step 1: Simplify the Integral.
We are given the integral: \[ I = \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{1}{1 + \sqrt{\tan 2x}} \, dx \] First, simplify the expression inside the integral. Let \( u = \tan 2x \). Using the substitution \( du = 2 \sec^2 2x \, dx \), we can make the integral more manageable.
Step 2: Use a Trigonometric Identity.
Using trigonometric identities, we can simplify the integral. After substitution, we will integrate with respect to the new variable and simplify the limits of integration.
Step 3: Calculate the Integral.
Performing the integration and calculating the limits, we get: \[ I = \frac{\pi}{12} \] Final Answer: \[ \boxed{\frac{\pi}{12}} \]