For the function y = 2x ^ 2-2x-5, find the antiderivative whose graph passes through point A (2; -1).

In general terms, the antiderivative equation for the function f (x) looks like this:

F (x) = ∫f (x) * dx + C, where C is a constant.

In this case, we get:

F (x) = ∫ (2x ^ 2 – 2x – 5) * dx + C = 2/3 * x ^ 3 – x ^ 2 – 5x + C.

Substitute the coordinates of point A into the resulting equation and find the value of C:

2/3 * 2 ^ 3 – 2 ^ 2 – 5 * 2 – C = -1;

16/3 – 13 – C = -1;

C = 16/3 – 39/3 = -23/3.

Answer: the desired antiderivative has the following form: 2 / 3x ^ 3 -x ^ 2 – 5x – 23/3.