To find the antiderivative of the function $\frac{7x^2 + 9x^4}{x^5}$, let's first simplify the expression by dividing each term in the numerator by $x^5$:

$\frac{7x^2}{x^5} + \frac{9x^4}{x^5} = 7x^{-3} + 9x^{-1}$

Now, we need to find the antiderivative of each term separately.

1. Antiderivative of $7x^{-3}$: $\int 7x^{-3} \, dx = 7 \int x^{-3} \, dx = 7 \cdot \frac{x^{-2}}{-2} = -\frac{7}{2} x^{-2}$

2. Antiderivative of $9x^{-1}$: $\int 9x^{-1} \, dx = 9 \ln |x|$

Combining these results, the antiderivative is: $-\frac{7}{2} x^{-2} + 9 \ln |x| + C$

So the final answer is: $\text{Antiderivative} = -\frac{7}{2x^2} + 9 \ln |x| + C$

Would you like a step-by-step breakdown of any part of the solution?

Here are some related questions for further practice:

1. What is the antiderivative of $\frac{5x^3 + 4x^2}{x^4}$?
2. How do you find the antiderivative of $x^{-3}$?
3. What are the steps to integrate $x^n$ where $n \neq -1$?
4. How would you approach integrating $\frac{3x^4 + 2x^2}{x^5}$?
5. Why is the constant of integration $C$ necessary in indefinite integrals?

Tip: When simplifying fractions with exponents, remember to subtract the exponent of the denominator from the exponent of the numerator for each term.