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

Answer

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

### Step 1: Identify the integral
We need to evaluate the definite integral of the function \( x \) from 0 to 1.

### Step 2: Find the antiderivative
The antiderivative of \( x \) can be calculated using the power rule of integration. The power rule states that:

\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

For our function \( x \), we can consider it as \( x^1 \). Thus, applying the power rule:

\[ \int x \, dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2} \]

### Step 3: Evaluate the definite integral
Now, we will evaluate the definite integral from 0 to 1 using the antiderivative we found:

\[ \int_{0}^{1} x \, dx = \left[ \frac{x^2}{2} \right]_{0}^{1} \]

### Step 4: Substitute the limits
We will substitute the upper limit (1) and the lower limit (0) into the antiderivative:

1. Substitute \( x = 1 \):
\[ \frac{1^2}{2} = \frac{1}{2} \]

2. Substitute \( x = 0 \):
\[ \frac{0^2}{2} = 0 \]

### Step 5: Calculate the result
Now, we will subtract the value at the lower limit from the value at the upper limit:

\[ \int_{0}^{1} x \, dx = \frac{1}{2} - 0 = \frac{1}{2} \]

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

\[ \frac{1}{2} \]

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