https://file.aiquickdraw.com/mathbot/file/1724738328640m29on0os.jpg

The image shows the integral:

\[
\int x^5 (2x^2 + 7x - 4) \, dx
\]

To solve this integral, we will first distribute \(x^5\) across the terms inside the parentheses and then integrate each term individually.

### Step-by-Step Solution

1. **Expand the integral:**

\[
\int x^5 (2x^2 + 7x - 4) \, dx = \int (2x^7 + 7x^6 - 4x^5) \, dx
\]

2. **Integrate each term separately:**

\[
\int 2x^7 \, dx + \int 7x^6 \, dx - \int 4x^5 \, dx
\]

3. **Apply the power rule of integration:**

The power rule states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) where \(C\) is the constant of integration.

- For \(\int 2x^7 \, dx\):

\[
\int 2x^7 \, dx = 2 \cdot \frac{x^{8}}{8} = \frac{x^8}{4}
\]

- For \(\int 7x^6 \, dx\):

\[
\int 7x^6 \, dx = 7 \cdot \frac{x^7}{7} = x^7
\]

- For \(\int 4x^5 \, dx\):

\[
\int 4x^5 \, dx = 4 \cdot \frac{x^6}{6} = \frac{2x^6}{3}
\]

4. **Combine the results:**

\[
\int x^5 (2x^2 + 7x - 4) \, dx = \frac{x^8}{4} + x^7 - \frac{2x^6}{3} + C
\]

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

### Final Answer

\[
\boxed{\frac{x^8}{4} + x^7 - \frac{2x^6}{3} + C}
\]

Would you like more details on any of the steps, or do you have any questions?

### Related Questions:

1. How do you apply the power rule for integration with negative exponents?
2. What is the process for integrating polynomials of higher degrees?
3. How does substitution work for integrals, and when should it be used?
4. Can you demonstrate integrating a polynomial using definite integrals?
5. How does one determine the constant of integration in practical applications?

**Tip:** When integrating polynomials, always break down the expression into simpler terms if possible. This makes it easier to apply the power rule and ensures accuracy.

Learn how to solve the integral of x^5 (2x^2 + 7x - 4) with a detailed step-by-step solution. This problem involves integrating polynomial functions using the power rule. Suitable for advanced high school students and college-level calculus enthusiasts.

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