AP Calculus AB Exam Question:

Evaluate the following definite integral:

∫(4x^3 + 2x^2 - 5x + 3) dx from x = 0 to x = 2.

Step-by-step explanation:

To evaluate a definite integral, we need to find the antiderivative of the given function and then substitute the upper and lower limits of integration. Let's go through the solution step-by-step:

Step 1: Find the antiderivative of the function. Using the power rule of integration, we find: ∫(4x^3 + 2x^2 - 5x + 3) dx = (4/4)x^4 + (2/3)x^3 - (5/2)x^2 + 3x + C, where C is the constant of integration.

Step 2: Plug in the upper and lower limits of integration. Substituting the upper limit (x = 2) and the lower limit (x = 0) into the antiderivative, we have: [(4/4)(2)^4 + (2/3)(2)^3 - (5/2)(2)^2 + 3(2)] - [(4/4)(0)^4 + (2/3)(0)^3 - (5/2)(0)^2 + 3(0)].

Simplifying this expression, we get: [(4/4)(16) + (2/3)(8) - (5/2)(4) + 6] - [0 + 0 - 0 + 0].

Step 3: Simplify the expression. [(4)(4) + (16/3) - (10) + 6] - [0 + 0 - 0 + 0] = 16 + 16/3 - 10 + 6.

Step 4: Combine like terms. 16 - 10 + 6 + 16/3 = 8 + 16/3.

Step 5: Convert the mixed number to an improper fraction. 8 + 16/3 = (24/3) + (16/3) = 40/3.

Therefore, the value of the definite integral ∫(4x^3 + 2x^2 - 5x + 3) dx from x = 0 to x = 2 is 40/3.