Evaluate the integral ∫(1 to 3) (3x^2 - 2) dx. (2019)

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Question: Evaluate the integral ∫(1 to 3) (3x^2 - 2) dx. (2019)

Options:

Correct Answer: 12

Exam Year: 2019

Solution:

∫(1 to 3) (3x^2 - 2) dx = [x^3 - 2x] from 1 to 3 = (27 - 6) - (1 - 2) = 20.

Evaluate the integral ∫(1 to 3) (3x^2 - 2) dx. (2019)

Evaluate the integral ∫(1 to 3) (3x^2 - 2) dx. (2019)

Step 1: Identify the integral to evaluate: ∫(1 to 3) (3x^2 - 2) dx.
Step 2: Find the antiderivative of the function 3x^2 - 2. The antiderivative is x^3 - 2x.
Step 3: Evaluate the antiderivative at the upper limit (3): (3^3 - 2*3). Calculate 3^3 = 27 and 2*3 = 6, so this gives 27 - 6 = 21.
Step 4: Evaluate the antiderivative at the lower limit (1): (1^3 - 2*1). Calculate 1^3 = 1 and 2*1 = 2, so this gives 1 - 2 = -1.
Step 5: Subtract the value at the lower limit from the value at the upper limit: 21 - (-1) = 21 + 1 = 22.
Step 6: The final answer is 22.

Definite Integral Evaluation – The question tests the ability to evaluate a definite integral using the Fundamental Theorem of Calculus.
Polynomial Integration – It involves integrating a polynomial function, specifically a quadratic function in this case.