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

Answer

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

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

### Step 2: Rewrite the Integral
Substituting \( t = \log x \) into the integral, we have:
\[ I = \int \frac{\sin(t)}{e^t} \cdot e^t \, dt \]
The \( e^t \) in the numerator and denominator cancels out:
\[ I = \int \sin(t) \, dt \]

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

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

### 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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