If $$\int \frac{dx}{x+x^7} = p(x)$$ then, $$\int \frac{x^6}{x+x^7}dx$$ is equal to:

We are given that $$\int \frac{dx}{x + x^7} = p(x)$$. We need to find $$\int \frac{x^6}{x + x^7} dx$$.

First, simplify the denominator in both integrals. Notice that $$x + x^7 = x(1 + x^6)$$. So, the given integral becomes:

$$p(x) = \int \frac{dx}{x(1 + x^6)}$$

Now, consider the integral we need to compute:

$$\int \frac{x^6}{x + x^7} dx = \int \frac{x^6}{x(1 + x^6)} dx$$

Simplify the fraction:

$$\frac{x^6}{x(1 + x^6)} = \frac{x^6}{x \cdot (1 + x^6)} = \frac{x^5}{1 + x^6}$$

So, the integral becomes:

$$\int \frac{x^5}{1 + x^6} dx$$

To relate this to $$p(x)$$, observe the expression for $$p(x)$$:

$$p(x) = \int \frac{dx}{x(1 + x^6)}$$

Consider the algebraic identity:

$$\frac{1}{x(1 + x^6)} = \frac{1}{x} - \frac{x^5}{1 + x^6}$$

Verify this by combining the terms on the right:

$$\frac{1}{x} - \frac{x^5}{1 + x^6} = \frac{1 + x^6 - x^6}{x(1 + x^6)} = \frac{1}{x(1 + x^6)}$$

This confirms the identity. Therefore, we can write:

$$p(x) = \int \left( \frac{1}{x} - \frac{x^5}{1 + x^6} \right) dx$$

Split the integral:

$$p(x) = \int \frac{1}{x} dx - \int \frac{x^5}{1 + x^6} dx$$

We know that $$\int \frac{1}{x} dx = \ln|x| + c_1$$, where $$c_1$$ is a constant. Let $$I = \int \frac{x^5}{1 + x^6} dx$$, so:

$$p(x) = \ln|x| - I + c_1$$

Solve for $$I$$:

$$I = \ln|x| - p(x) + c_1$$

But $$I$$ is exactly the integral we need, $$\int \frac{x^5}{1 + x^6} dx = \int \frac{x^6}{x + x^7} dx$$. Therefore:

$$\int \frac{x^6}{x + x^7} dx = \ln|x| - p(x) + c$$

where $$c = c_1$$ is the constant of integration.

Comparing with the options:

A. $$\ln|x| - p(x) + c$$

B. $$\ln|x| + p(x) + c$$

C. $$x - p(x) + c$$

D. $$x + p(x) + c$$

Option A matches our result.

Hence, the correct answer is Option A.