Find the antiderivative of a function whose graph passes through the point M (-2; 1) F (x) = x ^ 2 + 6x + 8

We have the function f (x) = x ^ 2 + 6 * x + 8.

Let’s find the antiderivative, the graph of which passes through the point (-2; 1).

We find the antiderivative piece by piece.

The antiderivative of the first term is a variable in the third degree, multiplied by a numerical coefficient:

F1 (x) = x ^ 3 * 1/3;

The antiderivative of the second term is a variable of the second degree multiplied by a numerical coefficient.

F2 (x) = x ^ 2 * 3;

The antiderivative of the third term is the product of a number and a variable x.

F3 (x) = 8 * x.

We connect the functions:

F (x) = 1/3 * x ^ 3 + 3 * x ^ 2 + 8 * x + C, where C is const.

Substitute the coordinate values:

1 = 1/3 * (-8) + 3 * 4 – 2 * 8 + C;

1 = -8/3 – 4 + C;

C – 7 2/3 = 0;

C = 7 2/3;

Our function: y = 1/3 * x ^ 3 + 3 * x ^ 2 + 8 * x + 7 2/3.