compute the integral :
`int_(1)^(2) (dx)/(x)`

Answer

To compute the integral \( \int_{1}^{2} \frac{dx}{x} \), we will follow these steps:

### Step 1: Identify the Integral
We need to evaluate the definite integral:
\[ \int_{1}^{2} \frac{dx}{x} \]

### Step 2: Find the Antiderivative
The antiderivative of \( \frac{1}{x} \) is:
\[ \int \frac{dx}{x} = \log |x| + C \]
where \( C \) is the constant of integration.

### Step 3: Apply the Limits
Now, we will evaluate the definite integral from 1 to 2:
\[ \left[ \log |x| \right]_{1}^{2} \]
This means we will substitute the upper limit (2) and the lower limit (1) into the antiderivative.

### Step 4: Substitute the Upper Limit
Substituting the upper limit \( x = 2 \):
\[ \log |2| = \log 2 \]

### Step 5: Substitute the Lower Limit
Substituting the lower limit \( x = 1 \):
\[ \log |1| = \log 1 \]
Since \( \log 1 = 0 \), we have:
\[ \log 1 = 0 \]

### Step 6: Calculate the Final Result
Now, we subtract the lower limit result from the upper limit result:
\[ \log 2 - \log 1 = \log 2 - 0 = \log 2 \]

Thus, the value of the integral is:
\[ \int_{1}^{2} \frac{dx}{x} = \log 2 \]

### Final Answer
\[ \log 2 \]

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