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To find the antiderivative of the function \(\frac{3x^2 + 2x^4}{x^5}\) in the variable \(x\) where \(x \neq 0\), we will start by simplifying the given expression and then integrate the simplified form.

1. Simplify the expression:
\[
\frac{3x^2 + 2x^4}{x^5} = \frac{3x^2}{x^5} + \frac{2x^4}{x^5} = 3x^{-3} + 2x^{-1}
\]

2. Integrate each term separately:
\[
\int (3x^{-3} + 2x^{-1}) \, dx
\]

3. Apply the power rule for integration:
   - For \( \int 3x^{-3} \, dx \):
\[
3 \int x^{-3} \, dx = 3 \cdot \frac{x^{-2}}{-2} = -\frac{3}{2} x^{-2}
\]
   - For \( \int 2x^{-1} \, dx \):
\[
2 \int x^{-1} \, dx = 2 \ln |x|
\]

4. Combine the results and include the constant of integration \(C\):
\[
\int \left( 3x^{-3} + 2x^{-1} \right) \, dx = -\frac{3}{2} x^{-2} + 2 \ln |x| + C
\]

So, the antiderivative of \(\frac{3x^2 + 2x^4}{x^5}\) is:
\[
-\frac{3}{2} x^{-2} + 2 \ln |x| + C
\]

Would you like further details or have any questions about this solution?

Here are five related questions:
1. How do you integrate a function of the form \(x^n\) where \(n \neq -1\)?
2. What is the integral of \(x^{-3}\)?
3. How do you handle the integral of \(x^{-1}\)?
4. Can you provide an example of integrating a polynomial function?
5. What is the importance of the constant of integration \(C\) in indefinite integrals?

**Tip:** When integrating functions that include \(x^{-1}\), remember that \(\int x^{-1} \, dx = \ln |x| + C\), which differs from the general power rule for integration.

Find an antiderivative of \(\frac{3x^2 + 2x^4}{x^5}\) in the variable \(x\) where \(x \neq 0\). Remember to include a '+ C' if appropriate.

Learn how to find the antiderivative of the expression (3x^2 + 2x^4)/x^5 with a step-by-step solution. This problem involves key calculus concepts like the power rule for integration and the natural logarithm. Suitable for students in Grades 11-12 or college-level calculus courses.

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