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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.

Find an antiderivative of \(\frac{7x^2 + 9x^4}{x^5}\) in the variable \(x\) where \(x \neq 0\). Remember to include a "+C" if appropriate.

Learn how to find the antiderivative of the function \(\frac{7x^2 + 9x^4}{x^5}\) using basic rules of calculus, including the power rule for antiderivatives. Step-by-step solution provided with integration constant.

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