`int_(1)^(3)(1)/(x)dx`

Answer

To solve the integral \( \int_{1}^{3} \frac{1}{x} \, dx \), we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_{1}^{3} \frac{1}{x} \, dx \]

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

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

### Step 4: Substitute the limits
Substituting the upper limit:
\[ \log |3| = \log 3 \]
Substituting the lower limit:
\[ \log |1| = \log 1 \]

### Step 5: Calculate the values
We know that:
\[ \log 1 = 0 \]
Thus, we can now compute:
\[ I = \log 3 - \log 1 = \log 3 - 0 = \log 3 \]

### Final Answer
Therefore, the value of the integral is:
\[ \int_{1}^{3} \frac{1}{x} \, dx = \log 3 \]

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