If `int(dx)/(x+x^7)=p(x)," then "intx^6/(x+x^7)dx` is equal to

Options

A. `log|x|-p(x)+C`
B. `log|x|+p(x)+C`
C. `x-p(x)+C`
D. `x+p(x)+C`

Answer

To solve the problem, we need to evaluate the integral:

\[ \int \frac{x^6}{x + x^7} \, dx \]

Given that:

\[ \int \frac{dx}{x + x^7} = P(x) \]

### Step-by-Step Solution:

1. **Rewrite the Integral:**
We can rewrite the integrand by adding and subtracting 1 in the numerator:

\[ \int \frac{x^6}{x + x^7} \, dx = \int \frac{x^6 + 1 - 1}{x + x^7} \, dx = \int \frac{x^6 + 1}{x + x^7} \, dx - \int \frac{1}{x + x^7} \, dx \]

2. **Simplify the First Integral:**
We can simplify the first integral:

\[ \int \frac{x^6 + 1}{x + x^7} \, dx = \int \frac{1 + x^6}{x(1 + x^6)} \, dx \]

This can be separated into two parts:

\[ = \int \frac{1}{x} \, dx + \int \frac{x^6}{x + x^7} \, dx \]

3. **Evaluate the First Integral:**
The first integral is straightforward:

\[ \int \frac{1}{x} \, dx = \ln |x| + C_1 \]

4. **Substitute Back:**
Now, we substitute back into our expression:

\[ \int \frac{x^6}{x + x^7} \, dx = \ln |x| + C_1 - P(x) \]

5. **Final Expression:**
Thus, we can conclude that:

\[ \int \frac{x^6}{x + x^7} \, dx = \ln |x| - P(x) + C \]

Where \( C \) is the constant of integration.