The given expression for which we need to find an antiderivative is:

$\frac{7x^4 + 6x^6}{x^7}$

First, simplify the expression by dividing each term in the numerator by $x^7$:

$= \frac{7x^4}{x^7} + \frac{6x^6}{x^7}$

This simplifies to:

$= 7x^{4-7} + 6x^{6-7} = 7x^{-3} + 6x^{-1}$

Now, we find the antiderivative of each term with respect to $x$.

Step 1: Antiderivative of $7x^{-3}$

The general rule for the antiderivative of $x^n$ is:

$\int x^n dx = \frac{x^{n+1}}{n+1}, \text{ for } n \neq -1$

So, applying this to $7x^{-3}$:

$\int 7x^{-3} dx = 7 \cdot \frac{x^{-3+1}}{-3+1} = 7 \cdot \frac{x^{-2}}{-2} = -\frac{7}{2}x^{-2}$

Step 2: Antiderivative of $6x^{-1}$

The antiderivative of $x^{-1}$ is:

$\int x^{-1} dx = \ln|x|$

So, applying this to $6x^{-1}$:

$\int 6x^{-1} dx = 6 \ln|x|$

Final Solution

Thus, the antiderivative of $\frac{7x^4 + 6x^6}{x^7}$ is:

$-\frac{7}{2}x^{-2} + 6 \ln|x| + C$

where $C$ is the constant of integration.

Would you like more details or explanations on any of the steps?

Here are 5 follow-up questions you might find helpful:

1. How do you calculate the antiderivative of $x^n$ for different values of $n$?
2. What happens if the exponent is $n = -1$, and why is the natural logarithm involved?
3. What is the difference between an antiderivative and a derivative?
4. How does the constant of integration affect the solution?
5. Can you explain the importance of the absolute value in $\ln|x|$?

Tip: Always check whether you need absolute values in logarithmic terms when integrating expressions that involve $x^{-1}$.