`int (1)/(x sin^(2) (log x))dx`

Answer

To solve the integral \( I = \int \frac{1}{x \sin^2(\log x)} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = \log x \). Then, differentiating both sides gives us:
\[ dt = \frac{1}{x} \, dx \quad \Rightarrow \quad dx = x \, dt = e^t \, dt \]
Since \( x = e^t \), we can rewrite the integral in terms of \( t \).

### Step 2: Rewrite the Integral
Substituting \( x \) and \( dx \) into the integral, we have:
\[ I = \int \frac{1}{e^t \sin^2(t)} e^t \, dt = \int \frac{1}{\sin^2(t)} \, dt \]

### Step 3: Simplify the Integral
The integral simplifies to:
\[ I = \int \csc^2(t) \, dt \]

### Step 4: Integrate
We know that the integral of \( \csc^2(t) \) is:
\[ \int \csc^2(t) \, dt = -\cot(t) + C \]

### Step 5: Back Substitute
Now, we substitute back \( t = \log x \):
\[ I = -\cot(\log x) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{1}{x \sin^2(\log x)} \, dx = -\cot(\log x) + C \]

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