Question

Найти первообразную функции f(x)=3x^2+4, проходящую через точку M(1; 7).

Answer 1

Найдем множество первообразных для исходной функции:
F(x) = x^3 + 4x + C, где C - любое число. Нам нужна первообразная, проходящая через (1;7), поэтому:
7 = 1^3 + 4*1 + C
7 = 1 + 4 + C
7 = 5 + C
C = 2
Значит, искомая первообразная:
F(x) = x^3 + 4x + 2

Answer 2

Finding the Antiderivative of f(x) = 3x^2 + 4 Passing Through the Point M(1, 7)

To find the antiderivative of the function f(x) = 3x^2 + 4 that passes through the point M(1, 7), we can use the power rule for integration. The power rule states that the antiderivative of x^n is (1/(n+1))x^(n+1) + C, where C is the constant of integration.

In this case, we have f(x) = 3x^2 + 4. To find the antiderivative, we can apply the power rule to each term separately. The antiderivative of 3x^2 is (1/3)x^3, and the antiderivative of 4 is 4x. Adding these two antiderivatives together, we get:

F(x) = (1/3)x^3 + 4x + C,

where C is the constant of integration.

To determine the value of the constant of integration, we can use the fact that the antiderivative F(x) passes through the point M(1, 7). Substituting x = 1 and y = 7 into the equation for F(x), we can solve for C:

7 = (1/3)(1)^3 + 4(1) + C 7 = 1/3 + 4 + C 7 = 13/3 + C C = 7 - 13/3 C = 12/3 - 13/3 C = -1/3.

Therefore, the antiderivative of f(x) = 3x^2 + 4 that passes through the point M(1, 7) is:

F(x) = (1/3)x^3 + 4x - 1/3.

Please let me know if there's anything else I can help you with!