The question involves evaluating the following integral:

$I = \int_0^1 x^6 (1-x)^7 \, dx$

This integral is in the form of the Beta function $B(m+1, n+1)$, which is defined as:

$B(p, q) = \int_0^1 t^{p-1} (1-t)^{q-1} \, dt, \quad \text{where } p, q > 0.$

Step-by-Step Solution:

1. Compare the given integral with the Beta function definition: $\int_0^1 x^{6} (1-x)^{7} \, dx.$ Here:

   • $x^6$ corresponds to $t^{p-1}$, so $p-1 = 6 \implies p = 7$,
   • $(1-x)^7$ corresponds to $(1-t)^{q-1}$, so $q-1 = 7 \implies q = 8$.

2. Therefore, the integral becomes: $I = B(7, 8).$

Final Answer:

The correct choice is:

(C) $B(7, 6)$