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The integral provided is:

\[
\int \left( \frac{5}{2}x^5 + x^3 - 2x^2 + \frac{3}{x} \right) dx
\]

To solve this, we can integrate each term separately:

1. \(\int \frac{5}{2} x^5 \, dx = \frac{5}{2} \cdot \frac{x^6}{6} = \frac{5}{12} x^6\)
2. \(\int x^3 \, dx = \frac{x^4}{4}\)
3. \(\int -2x^2 \, dx = -2 \cdot \frac{x^3}{3} = -\frac{2}{3} x^3\)
4. \(\int \frac{3}{x} \, dx = 3 \ln |x|\)

Now, summing all these together:

\[
\frac{5}{12} x^6 + \frac{x^4}{4} - \frac{2}{3} x^3 + 3 \ln |x| + C
\]

Where \(C\) is the constant of integration.

Would you like further explanation on any of these steps?

Here are 5 related questions to think about:
1. How do we handle integrals with logarithmic terms like \(\frac{3}{x}\)?
2. What are the general rules for integrating polynomials?
3. Can you apply similar steps to integrate \(\int (x^4 - x^2 + \frac{1}{x^2}) dx\)?
4. How does the power rule for integration work for negative powers of \(x\)?
5. How would the result change if one of the terms had a fractional exponent?

**Tip**: When integrating, it's important to carefully apply the power rule for different powers of \(x\), especially for terms like \(\frac{1}{x}\), which lead to logarithmic integrals.

Learn how to solve the integral ∫[(5/2)x^5 + x^3 - 2x^2 + (3/x)] dx with a detailed step-by-step explanation. This problem involves key calculus concepts like polynomial and logarithmic integration, suitable for college-level students in Calculus I.

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