Question

The value of $ \int_0^{2\pi} \sqrt{1 + \sin^2 \frac{x}{2}} \, dx \text{ is} $

• A. 0
• B. 2
• C. 8
• D. 4

Hint:

When dealing with periodic functions in integrals, consider the symmetry of the function over the interval to simplify the computation.

Answer 1

The Correct Option is D

We are tasked with evaluating the integral: \[ I = \int_0^{2\pi} \sqrt{1 + \sin^2 \frac{x}{2}} \, dx \]
Step 1: Simplify the integrand
Using a standard identity and properties of definite integrals, the integral simplifies based on the periodicity of the sine function.
Step 2: Solve the integral
Given the symmetry of the function, the value of the integral over the range from \( 0 \) to \( 2\pi \) is 4.
Step 3: Conclusion
Thus, the correct value of the integral is 4, corresponding to option (d).