`int (f'(x))/( f(x) log(f(x)))dx` is equal to

Options

A. `f(x) log (f(x)) +C`
B. `log(log(f(x))) +C`
C. `(f(x))/(log(f(x)) ) +C`
D. `(1)/( log(log(f(x))))+C`

Answer

To solve the integral

\[ \int \frac{f'(x)}{f(x) \log(f(x))} \, dx, \]

we can use substitution. Let's go through the steps in detail.

### Step 1: Substitution
Let
\[ t = \log(f(x)). \]
Then, we differentiate \(t\) with respect to \(x\):

\[ dt = \frac{1}{f(x)} f'(x) \, dx. \]

This implies that

\[ f'(x) \, dx = f(x) \, dt. \]

### Step 2: Rewrite the Integral
Now, we can rewrite the integral using our substitution:

\[ \int \frac{f'(x)}{f(x) \log(f(x))} \, dx = \int \frac{f(x) \, dt}{f(x) \cdot t} = \int \frac{dt}{t}. \]

### Step 3: Integrate
The integral of \(\frac{1}{t}\) is:

\[ \int \frac{dt}{t} = \log |t| + C, \]

where \(C\) is the constant of integration.

### Step 4: Substitute Back
Now, we substitute back \(t = \log(f(x))\):

\[ \log |t| + C = \log |\log(f(x))| + C. \]

### Final Answer
Thus, the final answer is:

\[ \log |\log(f(x))| + C. \]