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.