AP Calculus AB Exam Question:

Find the antiderivative of the function f(x) = 3x^2 + 2x + 1.

Step-by-step detailed explanation:

To find the antiderivative of a function, we need to reverse the process of differentiation. This involves finding a function whose derivative is equal to the original function.

Given that f(x) = 3x^2 + 2x + 1, we can find its antiderivative by applying the power rule for integration. As the power rule states:

∫ x^n dx = (1/(n+1)) * x^(n+1) + C,

where ∫ represents the integration, n represents any real number except -1, and C is the constant of integration.

To find the antiderivative of f(x), we integrate each term separately:

• The antiderivative of 3x^2 is (3/3) * x^3 = x^3.
• The antiderivative of 2x is (2/2) * x^2 = x^2.
• The antiderivative of 1 is x.

Adding up all the antiderivatives, we get the antiderivative of f(x) as:

∫ (3x^2 + 2x + 1) dx = x^3 + x^2 + x + C,

where C is the constant of integration.

Hence, the antiderivative of f(x) = 3x^2 + 2x + 1 is x^3 + x^2 + x + C.