The given function is \(\displaystyle \sqrt{1 + \sin \frac{x}{2}} \) and we need to compute the integral of this function with respect to \( x \), i.e., compute \(\displaystyle \int \sqrt{1 + \sin \frac{x}{2}} \, dx \).
Solution:
First, we simplify the expression inside the square root. We know that:
\( \begin{aligned} &1 + \sin \theta \\ &= \sin^2 \frac{\theta}{2} + \cos^2 \frac{\theta}{2} + 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \\ &= \left(\sin \frac{\theta}{2} + \cos \frac{\theta}{2}\right)^2 \\ \end{aligned} \)Let \(\displaystyle \theta = \frac{x}{2} \), then: \(\displaystyle 1 + \sin \frac{x}{2} = \left(\sin \frac{x}{4} + \cos \frac{x}{4}\right)^2 \)
Thus: \( \displaystyle \sqrt{1 + \sin \frac{x}{2}} = \sin \frac{x}{4} + \cos \frac{x}{4} \).
Therefore, the integral became:
\( \begin{aligned} &\int \sqrt{1 + \sin \frac{x}{2}} \, dx \\ &= \int \left(\sin \frac{x}{4} + \cos \frac{x}{4}\right) dx \\ &= \int \sin \frac{x}{4} \, dx + \int \cos \frac{x}{4} \, dx \\ &= -4 \cos \frac{x}{4} + 4 \sin \frac{x}{4} + C, \quad \text{C is a constant} \\ \end{aligned} \)Thus, the integral of the given function is
\( \boxed{\int \sqrt{1 + \sin \frac{x}{2}} \, dx = 4 \left(\sin \frac{x}{4} – \cos \frac{x}{4}\right) + C} \)Please let me know in the comments if you find any errors in this solution.