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!