`int(tan(logx))/(x) dx` =

Options

A. `"log cos" (log x)`
B. `"log sec "("log" x)`
C. `"log sin " ("log" x)`
D. None of these

Answer

To solve the integral \( \int \frac{\tan(\log x)}{x} \, dx \), we will use substitution. Here are the 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 rewrite \( dx \) as:
\[ dx = e^t \, dt \]

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

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

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

### Final Answer
Thus, the final answer is:
\[ -\log|\cos(\log x)| + C \]