`int("cosec"^(2)(logx))/(x)dx`

Answer

To solve the integral \( \int \frac{\csc^2(\log x)}{x} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = \log x \). Then, the differential \( dt \) can be found using the derivative of \( \log x \):
\[ dt = \frac{1}{x} \, dx \quad \Rightarrow \quad dx = x \, dt = e^t \, dt \]
Since \( x = e^t \), we can rewrite the integral in terms of \( t \).

### Step 2: Change of Variables
Now, substituting \( x \) and \( dx \) into the integral:
\[ \int \frac{\csc^2(\log x)}{x} \, dx = \int \csc^2(t) \, dt \]

### Step 3: Integrate
The integral of \( \csc^2(t) \) is a standard integral:
\[ \int \csc^2(t) \, dt = -\cot(t) + C \]

### Step 4: Back Substitute
Now, we substitute back \( t = \log x \):
\[ -\cot(t) + C = -\cot(\log x) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{\csc^2(\log x)}{x} \, dx = -\cot(\log x) + C \]

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