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.