Evaluate the following integrals :
`int ("In" x + (1)/( "In" x) ) (dx)/( x)`.

Answer

To evaluate the integral

\[ I = \int \left( \ln x + \frac{1}{\ln x} \right) \frac{dx}{x} \]

we can break this integral into two parts:

\[ I = \int \frac{\ln x}{x} \, dx + \int \frac{1}{\ln x} \cdot \frac{dx}{x} \]

### Step 1: Evaluate the first integral

For the first integral, we can use the substitution \( t = \ln x \). Then, we have:

\[ dt = \frac{1}{x} \, dx \quad \Rightarrow \quad dx = x \, dt = e^t \, dt \]

Substituting \( x = e^t \) into the integral gives:

\[ \int \frac{\ln x}{x} \, dx = \int t \, dt \]

Now, we can integrate \( t \):

\[ \int t \, dt = \frac{t^2}{2} + C = \frac{(\ln x)^2}{2} + C \]

### Step 2: Evaluate the second integral

Now, we evaluate the second integral:

\[ \int \frac{1}{\ln x} \cdot \frac{dx}{x} \]

Using the same substitution \( t = \ln x \), we have:

\[ \int \frac{1}{\ln x} \cdot \frac{dx}{x} = \int \frac{1}{t} \, dt \]

The integral of \( \frac{1}{t} \) is:

\[ \int \frac{1}{t} \, dt = \ln |t| + C = \ln |\ln x| + C \]

### Step 3: Combine the results

Now we can combine the results of both integrals:

\[ I = \frac{(\ln x)^2}{2} + \ln |\ln x| + C \]

### Final Answer

Thus, the final result of the integral is:

\[ \int \left( \ln x + \frac{1}{\ln x} \right) \frac{dx}{x} = \frac{(\ln x)^2}{2} + \ln |\ln x| + C \]