Question

Evaluate the integral \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt[3]{\sec x}}{\sqrt[3]{\sec x} + \sqrt[3]{\csc x}} \, dx \]

• A. 0
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{2} \)
• D. \( -\frac{\pi}{4} \)

Hint:

When simplifying integrals involving trigonometric functions, look for symmetry or substitutions to reduce the complexity.

Answer 1

The Correct Option is B

Step 1: Simplify the integrand.
We begin by recognizing the symmetry in the integral. By simplifying the expression using trigonometric identities, we can find that the integral evaluates to \( \frac{\pi}{4} \).
Step 2: Conclusion.
Thus, the value of the integral is \( \frac{\pi}{4} \), corresponding to option (B).