Find `int(x^(6)+x^(-6))dx`

Options

A. `(x^(6))/(7)-(x^(-5))/(5)`
B. `(x^(7))/(7)-(x^(-5))/(5)`
C. `(x^(7))/(7)-(x^(-5))`
D. `(x^(6))/(7)-(x^(-5))/(5)`

Explanation

To solve the integral \( \int (x^6 + x^{-6}) \, dx \), we can break it down into two separate integrals:

1. **Separate the integral:**
\[ \int (x^6 + x^{-6}) \, dx = \int x^6 \, dx + \int x^{-6} \, dx \]

2. **Integrate \( x^6 \):**
The formula for integrating \( x^n \) is:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
Applying this for \( n = 6 \):
\[ \int x^6 \, dx = \frac{x^{6+1}}{6+1} + C = \frac{x^7}{7} + C_1 \]

3. **Integrate \( x^{-6} \):**
Similarly, we apply the same formula for \( n = -6 \):
\[ \int x^{-6} \, dx = \frac{x^{-6+1}}{-6+1} + C = \frac{x^{-5}}{-5} + C_2 = -\frac{x^{-5}}{5} + C_2 \]

4. **Combine the results:**
Now, we combine the results of both integrals:
\[ \int (x^6 + x^{-6}) \, dx = \frac{x^7}{7} - \frac{x^{-5}}{5} + C \]

5. **Final answer:**
Thus, the final result of the integral is:
\[ \int (x^6 + x^{-6}) \, dx = \frac{x^7}{7} - \frac{1}{5x^5} + C \]