Write down the integral of the following
`x^4+x^-5`

Answer

To find the integral of the expression \( x^4 + x^{-5} \), we will follow the integration rules for each term separately.

### Step-by-Step Solution:

1. **Set up the integral**:
We start by writing the integral of the given expression:
\[ \int (x^4 + x^{-5}) \, dx \]

2. **Separate the integral**:
We can separate the integral into two parts:
\[ \int x^4 \, dx + \int x^{-5} \, dx \]

3. **Integrate the first term**:
For the first term \( \int x^4 \, dx \), we use the power rule of integration:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
Here, \( n = 4 \):
\[ \int x^4 \, dx = \frac{x^{4+1}}{4+1} = \frac{x^5}{5} \]

4. **Integrate the second term**:
Now, we integrate the second term \( \int x^{-5} \, dx \):
Here, \( n = -5 \):
\[ \int x^{-5} \, dx = \frac{x^{-5+1}}{-5+1} = \frac{x^{-4}}{-4} = -\frac{x^{-4}}{4} \]

5. **Combine the results**:
Now, we combine the results of both integrals:
\[ \int (x^4 + x^{-5}) \, dx = \frac{x^5}{5} - \frac{x^{-4}}{4} + C \]

6. **Final expression**:
We can express \( -\frac{x^{-4}}{4} \) in a different form if needed:
\[ -\frac{x^{-4}}{4} = -\frac{1}{4x^4} \]
Thus, the final answer can be written as:
\[ \int (x^4 + x^{-5}) \, dx = \frac{x^5}{5} - \frac{1}{4x^4} + C \]

### Final Answer:
\[ \int (x^4 + x^{-5}) \, dx = \frac{x^5}{5} - \frac{1}{4x^4} + C \]