\( \int \operatorname { ctg } x \sqrt { \ln | \sin x | } \)

To solve the integral \[ I = \int \cot x \sqrt{\ln |\sin x|} \, dx, \] we can use integration by parts. Let's set: - \( u = \sqrt{\ln |\sin x|} \) - \( dv = \cot x \, dx \) Next, we need to find \( du \) and \( v \). 1. **Finding \( du \)**: Using the chain rule, we have: \[ du = \frac{1}{2\sqrt{\ln |\sin x|}} \cdot \frac{1}{\sin x} \cdot \cos x \, dx = \frac{\cos x}{2\sin x \sqrt{\ln |\sin x|}} \, dx = \frac{1}{2 \tan x \sqrt{\ln |\sin x|}} \, dx. \] 2. **Finding \( v \)**: The integral of \( \cot x \) is: \[ v = \ln |\sin x|. \] Now we can apply integration by parts: \[ I = uv - \int v \, du. \] Substituting \( u \) and \( v \): \[ I = \sqrt{\ln |\sin x|} \ln |\sin x| - \int \ln |\sin x| \cdot \frac{1}{2 \tan x \sqrt{\ln |\sin x|}} \, dx. \] This integral is quite complex and does not simplify easily. Therefore, we can conclude that the integral \[ \int \cot x \sqrt{\ln |\sin x|} \, dx \] does not have a simple closed form in terms of elementary functions. Thus, the integral can be expressed as: \[ I = \sqrt{\ln |\sin x|} \ln |\sin x| - \frac{1}{2} \int \frac{\ln |\sin x|}{\tan x \sqrt{\ln |\sin x|}} \, dx + C, \] where \( C \) is the constant of integration. In summary, the integral does not yield a simple expression, and further numerical or special function methods may be required for specific evaluations.