Step-by-step Solution:

\[ \int \sqrt{1 + 2\cot x (\csc x + \cot x)} \, dx \quad \text{where } 0 < x < \frac{\pi}{2} \] We simplify the expression: \[ 1 + 2\cot x (\csc x + \cot x) \] Recall the identities: \[\] \(\cot x = \frac{\cos x}{\sin x}\)\[\] \(\csc x = \frac{1}{\sin x}\) \[\] So: \[ 2\cot x (\csc x + \cot x) = 2\cot x \csc x + 2\cot^2 x \] So the expression becomes: \[ 1 + 2\cot x \csc x + 2\cot^2 x \] Now try to write this in a perfect square form: Try: \[ ( \cot x + \csc x )^2 = \cot^2 x + \csc^2 x + 2\cot x \csc x \] Recall: \[ \csc^2 x = 1 + \cot^2 x \Rightarrow \cot^2 x + \csc^2 x = 1 + 2\cot^2 x \] So: \[ ( \cot x + \csc x )^2 = 1 + 2\cot^2 x + 2\cot x \csc x \] Exactly matches the expression under the root! \[\] Step 2: Take the square root. \[ \sqrt{1 + 2\cot x (\csc x + \cot x)} = \sqrt{( \cot x + \csc x )^2} = \cot x + \csc x \quad \text{(since all values are positive in } (0, \frac{\pi}{2})) \] Step 3: Integrate. \[ \int (\cot x + \csc x) \, dx = \int \cot x \, dx + \int \csc x \, dx \] \(\int \cot x \, dx = \ln|\sin x|\) \[\] \(\int \csc x \, dx = \ln|\csc x - \cot x|\) \[\] So the final result is: \[ \ln|\sin x| + \ln|\csc x - \cot x| + C = \ln \left( \sin x (\csc x - \cot x) \right) + C \] Now simplify \(\sin x (\csc x - \cot x)\): \[ \sin x \left( \frac{1}{\sin x} - \frac{\cos x}{\sin x} \right) = \sin x \cdot \frac{1 - \cos x}{\sin x} = 1 - \cos x \] \[ {\int \sqrt{1 + 2\cot x (\csc x + \cot x)} \, dx = \ln(1 - \cos x) + C = 2\log \left(\sin \tfrac{x}{2} \right) + C} \]