Evaluate the following integrals :
`int (x + 1/x)dx`
(a and b are constant)

Answer

To evaluate the integral \( \int (x + \frac{1}{x}) \, dx \), we can break it down into two separate integrals. Here’s the step-by-step solution:

### Step 1: Break down the integral
We can separate the integral into two parts:
\[ \int (x + \frac{1}{x}) \, dx = \int x \, dx + \int \frac{1}{x} \, dx \]

### Step 2: Evaluate the first integral
The first integral is \( \int x \, dx \). Using the power rule for integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) (where \( C \) is the constant of integration), we have:
\[ \int x \, dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2} \]

### Step 3: Evaluate the second integral
The second integral is \( \int \frac{1}{x} \, dx \). This integral is known to be:
\[ \int \frac{1}{x} \, dx = \ln |x| + C \]

### Step 4: Combine the results
Now, we can combine the results of the two integrals:
\[ \int (x + \frac{1}{x}) \, dx = \frac{x^2}{2} + \ln |x| + C \]

### Final Result
Thus, the final result of the integral is:
\[ \int (x + \frac{1}{x}) \, dx = \frac{x^2}{2} + \ln |x| + C \]