https://file.aiquickdraw.com/mathbot/file/1727431015494jgl29rw8.jpg

The question in the image is in Ukrainian and asks to find the general form of the **antiderivative** (or the indefinite integral) for the function \( f(x) = x^8 \).

### Solution:
To find the antiderivative of \( f(x) = x^8 \), we apply the power rule of integration. The power rule states that for any function of the form \( x^n \), the antiderivative is given by:

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]

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

#### Applying the rule to \( f(x) = x^8 \):
\[
\int x^8 \, dx = \frac{x^{8+1}}{8+1} + C = \frac{x^9}{9} + C
\]

### Final Answer:
The general form of the antiderivative is:

\[
F(x) = \frac{x^9}{9} + C
\]

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:
1. What is the antiderivative of \( f(x) = x^n \) for any integer \( n \)?
2. How do you compute definite integrals after finding the antiderivative?
3. What is the significance of the constant \( C \) in indefinite integrals?
4. Can you find the antiderivative of a polynomial function, such as \( f(x) = 3x^2 + 5x - 7 \)?
5. How would the process change if you were to integrate functions like \( x^{-8} \)?

**Tip**: When integrating, always remember to add the constant \( C \), as it represents the family of all antiderivatives.

Знайдіть загальний вигляд первісної для функції f(x) = x^8.

Learn how to find the antiderivative for the function f(x) = x^8 using the power rule of integration. This problem covers calculus concepts, including the application of the power rule, and is suitable for students in Grades 10-12.

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