It looks like you've uploaded an image showing the steps to find the antiderivative of the expression:

$\frac{7x^4 - 9x^6}{x^7}$

Let me help you walk through the solution for this.

Step 1: Simplify the expression

We first simplify the given expression:

$\frac{7x^4 - 9x^6}{x^7} = \frac{7x^4}{x^7} - \frac{9x^6}{x^7}$

This simplifies to:

$7x^{-3} - 9x^{-1}$

Step 2: Find the antiderivative of each term

Now, we find the antiderivative of each term separately.

1. The antiderivative of $7x^{-3}$ is:

$\frac{7}{-2}x^{-2} = -\frac{7}{2}x^{-2}$

2. The antiderivative of $-9x^{-1}$ is:

$-9 \ln|x|$

Step 3: Combine the results

Putting it all together, we get:

$-\frac{7}{2}x^{-2} - 9\ln|x| + C$

Final Result:

$\boxed{-\frac{7}{2}x^{-2} - 9\ln|x| + C}$

Let me know if you'd like further details or if you have any other questions!

Here are some related questions you could explore:

1. How do you find the antiderivative of rational functions?
2. What is the significance of the constant $C$ in antiderivatives?
3. How does logarithmic integration work for functions involving $x^{-1}$?
4. How can you verify the correctness of an antiderivative?
5. What are the rules for simplifying terms before integrating?

Tip: Always check if you can simplify the function first before taking the antiderivative to make the process easier!