To solve the indefinite integral

∫ (cos(2√x) / sin(x)) dx,
we can utilize substitution and trigonometric identities to simplify the expression.

Step 1: Substitute Variable
- **Let**: \( u = \sqrt{x} \)
- **Then**: \( x = u^2 \) and \( dx = 2u \, du \)
This changes the integral to:
- **Integral becomes**:
\( ∫ \frac{cos(2u)}{sin(u^2)} \cdot 2u \, du \)

Step 2: Use Trigonometric Identity
- Apply the double angle formula:
\( cos(2u) = 2cos^2(u) - 1 \)
- This gives:
\( ∫ \frac{(2cos^2(u) - 1) \cdot 2u}{sin(u^2)} \, du \)

Step 3: Break Down the Integral
- Split the integral:
\( 2∫ \frac{u \cdot cos^2(u)}{sin(u^2)} \, du - 2∫ \frac{u}{sin(u^2)} \, du \)

Step 4: Evaluate Each Integral
- The integrals \( ∫ \frac{u \cdot cos^2(u)}{sin(u^2)} \, du \) and \( ∫ \frac{u}{sin(u^2)} \, du \) can be evaluated using techniques like integration by parts or further substitutions.


Final Result
- Combine results and add the constant of integration \( C \):
\( ∫ (cos(2√x) / sin(x)) dx = \text{Result} + C \)
This integral may require numeric or computational methods for exact evaluations, but the outlined steps provide a clear path to simplification and analysis.