A

To solve the given integrals, we will integrate each one separately, using the fundamental rules of integration.

(i) For the integral:
$$ \int x^{11} \, dx $$
We apply the power rule of integration:
$$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $$
Here, n = 11. Thus,
$$ \int x^{11} \, dx = \frac{x^{12}}{12} + C_1 $$

(ii) For the integral:
$$ \int \left( \frac{1}{x^7} \right) \, dx $$
This can be rewritten as:
$$ \int x^{-7} \, dx $$
Applying the power rule again:
$$ \int x^{-7} \, dx = \frac{x^{-7+1}}{-7+1} + C = \frac{x^{-6}}{-6} + C_2 = -\frac{1}{6x^6} + C_2 $$

Therefore, the integrals together can be represented as:
$$ \int x^{11} \, dx + \int \left( \frac{1}{x^7} \right) \, dx = \frac{x^{12}}{12} - \frac{1}{6x^6} + C $$
Where C = C_1 + C_2, the constant of integration. Hence, the final answer is:
$$ \frac{x^{12}}{12} - \frac{1}{6x^6} + C $$
This corresponds to option A.