You provided an integral:

\[
\int \left( x^4 + 9x^2 + 5x^5 + 15x^3 + 25x \right) \, dx
\]

Let’s break it down and solve the integral term by term.

### 1. \( \int x^4 \, dx \):
Using the power rule: 
\[
\frac{x^{4+1}}{4+1} = \frac{x^5}{5}
\]

### 2. \( \int 9x^2 \, dx \):
Using the power rule and factoring out the constant 9:
\[
9 \cdot \frac{x^{2+1}}{2+1} = 9 \cdot \frac{x^3}{3} = 3x^3
\]

### 3. \( \int 5x^5 \, dx \):
Using the power rule and factoring out the constant 5:
\[
5 \cdot \frac{x^{5+1}}{5+1} = 5 \cdot \frac{x^6}{6} = \frac{5x^6}{6}
\]

### 4. \( \int 15x^3 \, dx \):
Using the power rule and factoring out the constant 15:
\[
15 \cdot \frac{x^{3+1}}{3+1} = 15 \cdot \frac{x^4}{4} = \frac{15x^4}{4}
\]

### 5. \( \int 25x \, dx \):
Using the power rule:
\[
25 \cdot \frac{x^{1+1}}{1+1} = 25 \cdot \frac{x^2}{2} = \frac{25x^2}{2}
\]

### Final Answer:
Now, summing up all the terms, the integral is:

\[
\int \left( x^4 + 9x^2 + 5x^5 + 15x^3 + 25x \right) \, dx = \frac{5x^6}{6} + \frac{15x^4}{4} + \frac{x^5}{5} + 3x^3 + \frac{25x^2}{2} + C
\]

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

Would you like further details on any part of this? Here are some related questions for practice:

1. How do you apply the power rule for integration?
2. What is the integral of \( x^n \) for any \( n \)?
3. How would you handle integrals with trigonometric functions?
4. Can you integrate functions involving logarithms?
5. What is the significance of the constant of integration?

**Tip**: Always check if terms can be simplified or factored before integrating—it can make the process smoother!

Learn how to solve the integral of a polynomial expression: x^4 + 9x^2 + 5x^5 + 15x^3 + 25x. This problem demonstrates the application of the power rule for integration, with detailed step-by-step solutions suitable for students in Grades 11-12 or early college level.

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