Question

The value of \( \int_0^{\frac{2\pi}{0}} \left( 1 + \sin \left( \frac{x}{2} \right) \right) \, dx \) is:

• A. 4
• B. 2
• C. 0
• D. 8

Hint:

When splitting an integral, remember to handle any trigonometric substitution carefully, and check the limits of integration.

Answer 1

The Correct Option is D

We are given the integral: \[ \int_0^{\frac{2\pi}{0}} \left( 1 + \sin \left( \frac{x}{2} \right) \right) \, dx \] We can split this into two integrals: \[ \int_0^{\frac{2\pi}{0}} 1 \, dx + \int_0^{\frac{2\pi}{0}} \sin \left( \frac{x}{2} \right) \, dx \] The first integral: \[ \int_0^{\frac{2\pi}{0}} 1 \, dx = \left[ x \right]_0^{\frac{2\pi}{0}} = \left( \frac{2\pi}{0} - 0 \right) = \frac{2\pi}{0} \] The second integral: Let \( u = \frac{x}{2} \), then \( du = \frac{dx}{2} \) or \( dx = 2du \). Thus, the second integral becomes: \[ \int_0^{\frac{2\pi}{4}} \sin \left( \frac{x}{2} \right) \, dx = 2 \int_0^{\frac{2\pi}{4}} \sin u \, du = 2 \left[ -\cos u \right]_0^{\frac{2\pi}{4}} = 2 \left( -\cos \left( \frac{2\pi}{4} \right) + \cos(0) \right) \] \[ = 2 \left( -\frac{\sqrt{2}}{2} + 1 \right) = 2 \left( 1 - \frac{\sqrt{2}}{2} \right) \] \[ = 2 \times \frac{2 - \sqrt{2}}{2} = 2 - \sqrt{2} \] Now adding both parts together: \[ \frac{2\pi}{0} + 2 - \sqrt{2} = 8 \] Thus, the correct answer is \( 8 \).