Find `int (sin(logx))/x dx`

Answer

To solve the integral \( \int \frac{\sin(\log x)}{x} \, dx \), we can use a substitution method. Here are the steps:

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

### Step 2: Change the Integral
Substituting \( t \) into the integral, we have:
\[ \int \frac{\sin(\log x)}{x} \, dx = \int \sin(t) \, dt \]

### Step 3: Integrate
Now we can integrate \( \sin(t) \):
\[ \int \sin(t) \, dt = -\cos(t) + C \]
where \( C \) is the constant of integration.

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

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

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