`int_(0)^(1) x dx`

Answer

To solve the definite integral \( \int_{0}^{1} x \, dx \), we can follow these steps:

### Step 1: Set up the integral
We start by writing the integral we want to evaluate:
\[ I = \int_{0}^{1} x \, dx \]

### 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 can apply the power rule:
\[ \int x \, dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2} \]

### Step 3: Evaluate the integral from 0 to 1
Now we need to evaluate the antiderivative from the limits 0 to 1:
\[ I = \left[ \frac{x^2}{2} \right]_{0}^{1} \]
This means we will substitute the upper limit (1) and the lower limit (0) into the antiderivative.

### Step 4: Substitute the limits
Substituting the upper limit:
\[ \frac{1^2}{2} = \frac{1}{2} \]
Substituting the lower limit:
\[ \frac{0^2}{2} = 0 \]

### Step 5: Calculate the definite integral
Now we subtract the value at the lower limit from the value at the upper limit:
\[ I = \frac{1}{2} - 0 = \frac{1}{2} \]

### Final Answer
Thus, the value of the definite integral \( \int_{0}^{1} x \, dx \) is:
\[ \frac{1}{2} \]

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