Evaluate `int (sin(log x))/(x)dx`

Answer

To evaluate the integral \( \int \frac{\sin(\log x)}{x} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = \log x \). Then, we differentiate both sides with respect to \( x \):
\[ \frac{dt}{dx} = \frac{1}{x} \implies dt = \frac{1}{x} \, dx \implies dx = x \, dt \]
Since \( x = e^t \), we can substitute \( dx \) in terms of \( dt \).

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

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

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

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

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