To find the antiderivative (also known as the primitive or the indefinite integral) of the function f(x) = 2x + 6x^2, we need to find a function F(x) such that F'(x) = f(x).
Let's find the antiderivative of f(x) by integrating each term individually: ∫2x dx = x^2 + C1, where C1 is the constant of integration ∫6x^2 dx = 2x^3 + C2, where C2 is the constant of integration
Therefore, the antiderivative of f(x) = 2x + 6x^2 is F(x) = x^2 + 2x^3 + C, where C is the constant of integration.
Given that N(1,5) is provided, we can find the value of C: F(1) = 1^2 + 2(1)^3 + C 5 = 1 + 2 + C C = 5 - 3 C = 2
Therefore, the antiderivative of f(x) = 2x + 6x^2 with the initial condition N(1,5) is F(x) = x^2 + 2x^3 + 2.
To find the antiderivative (also known as the primitive or the indefinite integral) of the function f(x) = 2x + 6x^2, we need to find a function F(x) such that F'(x) = f(x).
Let's find the antiderivative of f(x) by integrating each term individually:
∫2x dx = x^2 + C1, where C1 is the constant of integration
∫6x^2 dx = 2x^3 + C2, where C2 is the constant of integration
Therefore, the antiderivative of f(x) = 2x + 6x^2 is F(x) = x^2 + 2x^3 + C, where C is the constant of integration.
Given that N(1,5) is provided, we can find the value of C:
F(1) = 1^2 + 2(1)^3 + C
5 = 1 + 2 + C
C = 5 - 3
C = 2
Therefore, the antiderivative of f(x) = 2x + 6x^2 with the initial condition N(1,5) is F(x) = x^2 + 2x^3 + 2.