Final answer:
The most general antiderivative (indefinite integral) of the function f(x) = x(12x - 8) is 4x^3 - 4x^2 + C. Differentiating this antiderivative results in the original function, confirming its correctness.
Step-by-step explanation:
To find the most general antiderivative of the function f(x) = x(12x - 8), we first distribute the x across the terms in the parenthesis, giving us f(x) = 12x^2 - 8x. Next, we find the antiderivative (integral) of each term separately.
The antiderivative of 12x^2 with respect to x is 12/3 x^3, which simplifies to 4x^3. The antiderivative of -8x is -8/2 x^2, which simplifies to -4x^2.
So the most general antiderivative of the function, also known as the indefinite integral, is 4x^3 - 4x^2 + C, where C represents the constant of integration. To confirm this is correct, we can take the derivative of our result, which should yield the original function, f(x).
Differentiating 4x^3 gives 12x^2 and differentiating -4x^2 gives -8x. Adding them up, we do indeed get the original function, 12x^2 - 8x, confirming our antiderivative is correct.