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The problem is asking to find the antiderivative (or indefinite integral) of the function \( f(x) = 4x^3 - 2x + 3 \), where the graph passes through the point \( A(1; 8) \).

### Step-by-Step Solution:

1. **Find the antiderivative**:
   To find the antiderivative \( F(x) \) of \( f(x) = 4x^3 - 2x + 3 \), we apply the power rule of integration to each term:

   \[
   \int 4x^3 \, dx = x^4 + C_1
   \]
   \[
   \int -2x \, dx = -x^2 + C_2
   \]
   \[
   \int 3 \, dx = 3x + C_3
   \]

   Therefore, the general form of the antiderivative is:
   \[
   F(x) = x^4 - x^2 + 3x + C
   \]
   where \( C \) is the constant of integration.

2. **Use the point \( A(1; 8) \)**:
   Since the function passes through the point \( A(1; 8) \), we can use this information to find \( C \). Substituting \( x = 1 \) and \( F(1) = 8 \):

   \[
   8 = (1)^4 - (1)^2 + 3(1) + C
   \]
   Simplifying:
   \[
   8 = 1 - 1 + 3 + C
   \]
   \[
   8 = 3 + C
   \]
   \[
   C = 5
   \]

3. **Final Answer**:
   The specific antiderivative is:
   \[
   F(x) = x^4 - x^2 + 3x + 5
   \]

### Summary:
The antiderivative of \( f(x) = 4x^3 - 2x + 3 \), with the condition that it passes through the point \( A(1; 8) \), is \( F(x) = x^4 - x^2 + 3x + 5 \).

Would you like more details, or do you have any further questions?

### Related Questions:
1. How can we verify that the point \( A(1; 8) \) lies on the graph of \( F(x) \)?
2. How do you compute the definite integral of \( f(x) \) over an interval?
3. What are the general rules for finding antiderivatives of polynomial functions?
4. What is the relationship between derivatives and antiderivatives?
5. How does the constant \( C \) in antiderivatives affect the graph of a function?

### Tip:
When solving for the constant in an antiderivative, make sure to use the given point correctly by substituting both \( x \) and \( F(x) \) values!

Знайти первісну функції f(x) = 4x^3 - 2x + 3, графік якої проходить через задану точку A(1; 8).

Solve the problem of finding the antiderivative of f(x) = 4x^3 - 2x + 3, where the graph passes through the point A(1,8). This involves integrating the polynomial function and applying the given point to find the constant of integration.

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