Question

Найдите первообразную функции f(x)= 4хкуб+ 8х-2 , график которой проходит через точку А(1;3)

Answer 1

f(x)=4x^3+8x-2
F(x) = x^4 + 4x^2 - 2x + C
Найдем С, подставив координаты точки: 3=1+4-2+С; 3=3+C; C=0

Искомая первообразная: F(x) = x^4 + 4x^2 - 2x 

Answer 2

Finding the Antiderivative of the Function

To find the antiderivative of the function \( f(x) = 4x^3 + 8x - 2 \) that passes through the point A(1, 3), we can use the standard rules of integration.

The antiderivative of a function \( f(x) \) is denoted by \( F(x) \), and it represents the set of all antiderivatives of \( f(x) \). In this case, we need to find the antiderivative \( F(x) \) such that \( F'(x) = f(x) \) and \( F(1) = 3 \).

The antiderivative of \( 4x^3 \) is \( x^4 \), the antiderivative of \( 8x \) is \( 4x^2 \), and the antiderivative of \( -2 \) is \( -2x \). Therefore, the antiderivative of \( f(x) = 4x^3 + 8x - 2 \) is:

\[ F(x) = x^4 + 4x^2 - 2x + C \]

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

Finding the Value of the Constant

To find the value of the constant \( C \), we can use the given point A(1, 3). Substituting \( x = 1 \) and \( F(x) = 3 \) into the antiderivative equation, we get:

\[ 1^4 + 4(1)^2 - 2(1) + C = 3 \] \[ 1 + 4 - 2 + C = 3 \] \[ 3 + C = 3 \] \[ C = 0 \]

Final Antiderivative

Therefore, the antiderivative of the function \( f(x) = 4x^3 + 8x - 2 \) that passes through the point A(1, 3) is:

\[ F(x) = x^4 + 4x^2 - 2x \]

This is the final antiderivative of the given function.

Note: The constant of integration \( C \) was found to be 0 by using the given point A(1, 3) to solve for \( C \).