Найти первообразную для функции f (x) = 4x-x^3, проходящую через точку М (2; 1), H (-2:3) Найти первообразную для функции f (x) = 4x-6x^2+1, проходящую через точку М (0; 4), Н (1; 3)

1. f (x) = 4x - x^3;

F (x) = ∫f (x) dx = ∫ (4x - x^3) dx = 2x^2 - x^4/4 + C;

a) M (2; 1);

F (2) = 1; 2 * 2^2 - 2^4/4 + C = 1; 8 - 4 + C = 1; C = - 3; F (x) = 2x^2 - x^4/4 - 3.

b) H (-2; 3);

F (-2) = 3; 2 * (-2) ^2 - (-2) ^4/4 + C = 3; 8 - 4 + C = 3; C = - 1; F (x) = 2x^2 - x^4/4 - 1.

2. f (x) = 4x - 6x^2 + 1;

F (x) = ∫f (x) dx = ∫ (4x - 6x^2 + 1) dx = 2x^2 - 2x^3 + x + C;

a) M (0; 4);

F (0) = 4; C = 4; F (x) = 2x^2 - 2x^3 + x + 4.

b) H (1; 3)

F (1) = 3; 2 * 1^2 - 2 * 1^3 + 1 + C = 3; 2 - 2 + 1 + C = 3; C = 2; F (x) = 2x^2 - 2x^3 + x + 2.