Evaluate the integral ∫_1^2 (3x^2 - 2) dx.

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Question: Evaluate the integral ∫_1^2 (3x^2 - 2) dx.

Options:

Correct Answer: 1

Solution:

∫_1^2 (3x^2 - 2) dx = [x^3 - 2x] from 1 to 2 = (8 - 4) - (1 - 2) = 3.

Evaluate the integral ∫_1^2 (3x^2 - 2) dx.

Evaluate the integral ∫_1^2 (3x^2 - 2) dx.

Step 1: Identify the integral to evaluate: ∫_1^2 (3x^2 - 2) dx.
Step 2: Find the antiderivative of the function 3x^2 - 2. The antiderivative is x^3 - 2x.
Step 3: Write the expression for the definite integral using the antiderivative: [x^3 - 2x] from 1 to 2.
Step 4: Substitute the upper limit (2) into the antiderivative: (2^3 - 2*2) = (8 - 4) = 4.
Step 5: Substitute the lower limit (1) into the antiderivative: (1^3 - 2*1) = (1 - 2) = -1.
Step 6: Calculate the difference between the upper and lower limits: 4 - (-1) = 4 + 1 = 5.

Definite Integral – The process of calculating the area under a curve defined by a function over a specific interval.
Fundamental Theorem of Calculus – Relates differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.
Evaluation of Limits – The process of substituting the upper and lower limits into the antiderivative to find the definite integral's value.