Evaluate the following :
`int _(-5)^(5) " x dx"`

Answer

To evaluate the integral \(\int_{-5}^{5} x \, dx\), we can follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_{-5}^{5} x \, dx \]

### Step 2: Find the antiderivative
The antiderivative of \(x\) is:
\[ \frac{x^2}{2} \]

### Step 3: Apply the limits of integration
Now, we apply the limits from \(-5\) to \(5\):
\[ I = \left[ \frac{x^2}{2} \right]_{-5}^{5} \]

### Step 4: Substitute the upper limit
First, substitute the upper limit \(5\):
\[ \frac{5^2}{2} = \frac{25}{2} \]

### Step 5: Substitute the lower limit
Next, substitute the lower limit \(-5\):
\[ \frac{(-5)^2}{2} = \frac{25}{2} \]

### Step 6: Calculate the result
Now, we subtract the lower limit result from the upper limit result:
\[ I = \frac{25}{2} - \frac{25}{2} = 0 \]

### Conclusion
Thus, the value of the integral is:
\[ \int_{-5}^{5} x \, dx = 0 \]