Evaluate the following integrals :
(b) `int ( dx)/( x "In" x) ,`

Answer

To evaluate the integral \( \int \frac{dx}{x \ln x} \), we can use a substitution method. Here are the steps:

### Step 1: Substitution
Let \( t = \ln x \). Then, the derivative of \( t \) with respect to \( x \) is:
\[ \frac{dt}{dx} = \frac{1}{x} \implies dx = x \, dt \]
Since \( x = e^t \) (from the substitution \( t = \ln x \)), we can express \( dx \) in terms of \( t \):
\[ dx = e^t \, dt \]

### Step 2: Rewrite the Integral
Now, substitute \( x \) and \( dx \) in the integral:
\[ \int \frac{dx}{x \ln x} = \int \frac{e^t \, dt}{e^t \cdot t} = \int \frac{dt}{t} \]

### Step 3: Integrate
The integral \( \int \frac{dt}{t} \) is a standard integral:
\[ \int \frac{dt}{t} = \ln |t| + C \]

### Step 4: Back Substitute
Now, substitute back \( t = \ln x \):
\[ \ln |t| + C = \ln |\ln x| + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{dx}{x \ln x} = \ln |\ln x| + C \]