A

Step 1: Solve the first integral
We need to integrate \( \int x^{11} dx \).
Using the power rule for integration, which states that \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where \( n \neq -1 \), we have:
\[ \int x^{11} dx = \frac{x^{12}}{12} + C_1 \]

Step 2: Solve the second integral
Now, we need to integrate \( \int \frac{1}{x^7} dx \).
This can be rewritten as \( \int x^{-7} dx \). Applying the power rule:
\[ \int x^{-7} dx = \frac{x^{-6}}{-6} + C_2 = -\frac{x^{-6}}{6} + C_2 = -\frac{1}{6x^6} + C_2 \]

Step 3: Combine the results
Hence, the total integral will be:
\[ \int x^{11} dx + \int \frac{1}{x^7} dx = \frac{x^{12}}{12} - \frac{1}{6x^6} + C \]

Final Answer
The solution to the given integrals is:
\( \frac{x^{12}}{12} - \frac{1}{6x^6} + C \). Thus, the answer is option A.