`int x^(-6) " dx "`

Answer

To solve the integral \( \int x^{-6} \, dx \), we will follow the steps outlined below:

### Step-by-Step Solution:

1. **Identify the integral**:
We start with the integral we need to solve:
\[ I = \int x^{-6} \, dx \]

2. **Use the power rule for integration**:
The power rule states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( n \neq -1 \). Here, \( n = -6 \).

3. **Apply the power rule**:
We substitute \( n = -6 \) into the formula:
\[ I = \frac{x^{-6 + 1}}{-6 + 1} + C \]

4. **Simplify the exponent and denominator**:
Calculate \( -6 + 1 = -5 \):
\[ I = \frac{x^{-5}}{-5} + C \]

5. **Rewrite the expression**:
This can be simplified further:
\[ I = -\frac{1}{5} x^{-5} + C \]

6. **Final answer**:
Thus, the integral is:
\[ \int x^{-6} \, dx = -\frac{1}{5} x^{-5} + C \]