`int(cos(logx))/(x)dx`

Answer

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

### Step 1: Substitution
Let \( t = \log x \). Then, we differentiate both sides with respect to \( x \):
\[ \frac{dt}{dx} = \frac{1}{x} \]
This implies that:
\[ dx = x \, dt = e^t \, dt \]
Since \( x = e^t \), we can also express \( \frac{1}{x} \) as \( e^{-t} \).

### Step 2: Rewrite the Integral
Now we can rewrite the integral in terms of \( t \):
\[ \int \frac{\cos(\log x)}{x} \, dx = \int \cos(t) \cdot e^{-t} \, dt \]

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

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

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

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