`int " x dx "`

Options

A. `(x^(2))/(2)+c`
B. `(x^(2))+c`
C. `x+c`
D. `(x^(1))/(2)+c`

Answer

To solve the integral \( \int x \, dx \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Integral**:
We start with the integral:
\[ I = \int x \, dx \]

2. **Use the Power Rule for Integration**:
The power rule for integration states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( n \neq -1 \) and \( C \) is the integration constant. In our case, \( n = 1 \).

3. **Apply the Power Rule**:
Substitute \( n = 1 \) into the power rule:
\[ I = \frac{x^{1+1}}{1+1} + C = \frac{x^2}{2} + C \]

4. **Final Result**:
Therefore, the integral evaluates to:
\[ I = \frac{x^2}{2} + C \]