`int(cos(log x))/(x) dx` =

Options

A. `sin [log (logx)]`
B. `log [sin (log x)]`
C. `log[cos (log x)]`
D. none

Answer

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

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

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

### Step 3: Integrate
Now we can integrate \( \cos(t) \):
\[ \int \cos(t) \, dt = \sin(t) + C \]

### Step 4: Back Substitute
Now we need to substitute back \( t = \log x \):
\[ \sin(t) = \sin(\log x) \]
Thus, the integral becomes:
\[ \int \frac{\cos(\log x)}{x} \, dx = \sin(\log x) + C \]

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

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