To solve the integral $\int \sqrt{\sin x} \, dx$, we need to use a substitution method. Here’s the step-by-step process:

1. Substitution: Let $u = \sin x$. Then, $du = \cos x \, dx$.

2. Adjusting the Integral: We also note that $\cos x = \sqrt{1 - \sin^2 x} = \sqrt{1 - u^2}$.

   Therefore, the integral becomes: $\int \sqrt{u} \, \frac{du}{\sqrt{1 - u^2}}$

3. Simplification: The integral simplifies to: $\int \frac{\sqrt{u}}{\sqrt{1 - u^2}} \, du$

4. Trigonometric Substitution: Use the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$, but in this form it’s more complex. It’s often more practical to look up the standard result for this type of integral.

The result is: $\int \sqrt{\sin x} \, dx = \frac{\sqrt{2}}{2} \left( \sqrt{\sin x} \cdot \text{EllipticE}\left( \frac{\pi}{2}, \sin x \right) - \text{EllipticF}\left( \frac{\pi}{2}, \sin x \right) \right) + C$ where $\text{EllipticE}$ and $\text{EllipticF}$ are the incomplete elliptic integrals of the first and second kind, respectively.

Would you like more detailed steps or explanations on any part of this process?

Here are some additional questions you might find useful:

1. How do I solve integrals involving trigonometric functions?
2. Can you explain the method of trigonometric substitution?
3. What are elliptic integrals and how are they used in calculus?
4. How do I perform integration by parts?
5. Can you provide examples of integrals solved using different techniques?

Tip: When dealing with complex integrals, it’s useful to recognize standard forms and results, which can often simplify the process significantly.