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

To find the value of the integral \(\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{1}{1 + \sqrt{\tan 2x}} \, dx\), we can employ a clever transformation technique.

First, take the substitution \( u = \frac{\pi}{12} - x \), which implies \( du = -dx \). The limits of integration will change accordingly. For \( x = \frac{\pi}{24} \), \( u = \frac{\pi}{12} - \frac{\pi}{24} = \frac{\pi}{24} \). For \( x = \frac{5\pi}{24} \), \( u = \frac{\pi}{12} - \frac{5\pi}{24} = -\frac{\pi}{24} \).

Thus, the original integral can be rewritten as:

\(-\int_{-\frac{\pi}{24}}^{\frac{\pi}{24}} \frac{1}{1 + \sqrt{\tan\left( \pi/6 - 2u\right)}} \, du\).

Now, observe the function inside the integral. Using the identity for the complementary angle of tangent:

\(\tan\left( \frac{\pi}{3} - x \right) = \frac{1}{\tan x}\)

Apply this identity: \(\tan(\pi/6 - 2u) = \tan(\pi/3 - (2x)) = \frac{1}{\tan 2x}\).

Thus, we have: \(\sqrt{\tan(\pi/6 - 2u)} = \frac{1}{\sqrt{\tan 2x}}\).

Hence, \(\frac{1}{1 + \sqrt{\tan(\pi/6 - 2u)}} = \frac{1}{1 + \frac{1}{\sqrt{\tan 2x}}}\).

Therefore, \(\frac{1}{1 + \frac{1}{\sqrt{\tan 2x}}} = \frac{\sqrt{\tan 2x}}{1 + \sqrt{\tan 2x}}\).

This transformation leads to the integral being symmetric, with complementing values that cancel each other out over this interval. The entire expression symmetry-wise results in half the value twisting the very symmetry it started with.

Fundamentally, the solution evaluates to:

\(\frac{\pi}{12}\).

• Option Analysis:
• Option \(\frac{\pi}{12}\) is the correct answer due to the symmetry and transformations applied.
• The other options don't satisfy the given integration parameters.

Therefore, the correct answer is \(\frac{\pi}{12}\).