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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 \( x^{-1} \).

Find an antiderivative of (7x^4 + 6x^6)/x^7 in the variable x where x ≠ 0.

Learn how to find the antiderivative of the expression (7x^4 + 6x^6)/x^7. This calculus problem covers key topics such as algebraic simplification, power rule integration, and logarithmic integration. Suitable for advanced high school or undergraduate students.

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