The image contains a multiple-choice question related to indefinite integrals in Russian. Let's break it down and answer it:

Question: What is defined as the indefinite integral of a function $f(x)$?

Answer options:

1. The limit of the ratio of the change in the function $\Delta y$ to the change in the argument $\Delta x$ as $\Delta x$ approaches zero.
2. The antiderivative function $F(x)$ for the function $f(x)$.
3. The set of all antiderivatives $F(x) + C$, where $C$ is an arbitrary constant.
4. The change of the antiderivative when the argument changes from $x = a$ to $x = b$.

Correct Answer:

Option 3: The set of all antiderivatives $F(x) + C$, where $C$ is an arbitrary constant.

Explanation: An indefinite integral is defined as the family of all antiderivatives of a function. It includes a constant $C$ since differentiation of a constant yields zero. Hence, $F(x) + C$ covers all possible antiderivatives.

Would you like a deeper explanation or have any questions about integrals?

Related Questions:

1. What is the primary difference between definite and indefinite integrals?
2. How do you prove that a given function is an antiderivative of another?
3. What is the role of the constant of integration $C$ in indefinite integrals?
4. Can you explain why the first option in the choices is related to derivatives, not integrals?
5. How does the Fundamental Theorem of Calculus connect indefinite and definite integrals?

Tip:

When solving indefinite integrals, always remember to add the integration constant $C$.