`int_(2)^(4) x dx`

Answer

To solve the integral \( \int_{2}^{4} x \, dx \), we can follow these steps:

### Step 1: Identify the integral
We need to evaluate the definite integral:
\[ I = \int_{2}^{4} x \, dx \]

### Step 2: Use the power rule for integration
The power rule states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
For our integral, \( n = 1 \). Therefore, we have:
\[ \int x \, dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2} \]

### Step 3: Apply the limits of integration
Now we will evaluate the integral from 2 to 4:
\[ I = \left[ \frac{x^2}{2} \right]_{2}^{4} \]

### Step 4: Substitute the upper limit
First, substitute the upper limit (4):
\[ \frac{4^2}{2} = \frac{16}{2} = 8 \]

### Step 5: Substitute the lower limit
Now, substitute the lower limit (2):
\[ \frac{2^2}{2} = \frac{4}{2} = 2 \]

### Step 6: Calculate the definite integral
Now, subtract the lower limit result from the upper limit result:
\[ I = 8 - 2 = 6 \]

### Final Answer
Thus, the value of the integral \( \int_{2}^{4} x \, dx \) is:
\[ \boxed{6} \]

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