For the function f (x) = x-3x ^ 2, find the antiderivative passing through the point (0; 2).

The antiderivative to the given function will look like this:

F (x) = ∫ (x – 3x ^ 2) * dx + C, where C is a constant.

Let’s find the indefinite integral:

F (x) = 1/2 * x ^ 2 – 3 * (1/3) * x ^ 3 = x ^ 2/2 – x ^ 3.

Substitute the coordinates of the given point and calculate C:

1/2 * 0 ^ 2 – 0 ^ 3 + C = 2

C = 2.

Answer: the antiderivative looks like this: F (x) = 1/2 * x ^ 2 – x ^ 3 + 2.