https://r2.erweima.ai/mathbot/image/1725935211654716ddbt4.jpg

The image you provided shows the following integral:

\[
\int (5x^2 + 3x - 2) \, dx
\]

The given solution for this integral is:

\[
\frac{5}{3}x^3 + \frac{3}{2}x^2 - 2x + C
\]

Let's break it down step-by-step:

1. **Integral of \(5x^2\):**
   \[
   \int 5x^2 \, dx = \frac{5}{3}x^3
   \]
   (Using the power rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\)).

2. **Integral of \(3x\):**
   \[
   \int 3x \, dx = \frac{3}{2}x^2
   \]
   (Again, applying the power rule).

3. **Integral of \(-2\):**
   \[
   \int -2 \, dx = -2x
   \]
   (Since \(\int c \, dx = cx\), where \(c\) is a constant).

4. **Constant of Integration:**
   The constant \(C\) represents an arbitrary constant added to the solution because the integral is indefinite.

Thus, the complete solution matches the one provided in the image:

\[
\frac{5}{3}x^3 + \frac{3}{2}x^2 - 2x + C
\]

Would you like more details on any specific part of this integral?

Here are 5 related questions you might find helpful:
1. What is the power rule for integration and how is it applied?
2. How does the constant of integration \(C\) affect indefinite integrals?
3. How do you differentiate the result to verify the original function?
4. How are integrals related to the area under a curve?
5. What is the difference between definite and indefinite integrals?

**Tip:** Always remember to add the constant of integration \(C\) when calculating indefinite integrals.

Learn how to calculate the integral of the polynomial expression 5x^2 + 3x - 2 with a detailed step-by-step solution. This problem introduces key integration concepts, including the power rule of integration, and is suitable for high school or introductory college-level calculus students.

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