Question

Evaluate the integral: \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\sqrt{\tan x}}{1 + \sqrt{\tan x}} \, dx \]

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{5} \)
• C. \( \frac{\pi}{10} \)
• D. \( \frac{\pi}{20} \)
• E. \( \frac{\pi}{2} \)

Hint:

When dealing with integrals involving trigonometric functions like \( \tan x \), using substitution methods can help simplify the expression. Look for standard integral forms to speed up the process.

Answer 1

The Correct Option is D

We are given the integral: \[ I = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\sqrt{\tan x}}{1 + \sqrt{\tan x}} \, dx \] To simplify this integral, let us perform the substitution \( t = \tan x \). 
Therefore: \[ dt = \sec^2 x \, dx \quad {or} \quad dx = \frac{dt}{\sec^2 x} \] Now, the limits of integration change with the substitution. When \( x = \frac{\pi}{5} \), we get \( t = \tan \frac{\pi}{5} \). When \( x = \frac{3\pi}{10} \), we get \( t = \tan \frac{3\pi}{10} \). Thus, the integral becomes: \[ I = \int_{\tan \frac{\pi}{5}}^{\tan \frac{3\pi}{10}} \frac{\sqrt{t}}{1 + \sqrt{t}} \cdot \frac{dt}{1+t} \] This is a standard form of a trigonometric integral, and after evaluating the integral (using known integrals or a suitable technique), we get: \[ I = \frac{\pi}{20} \] Thus, the value of the integral is \( \frac{\pi}{20} \).
Thus, the correct answer is option (D), \( \frac{\pi}{20} \).