Calculate ∫ from 1 to 3 of (2x + 1) dx.

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Question: Calculate ∫ from 1 to 3 of (2x + 1) dx.

Options:

Correct Answer: 6

Solution:

The integral evaluates to [x^2 + x] from 1 to 3 = (9 + 3) - (1 + 1) = 10.

Calculate ∫ from 1 to 3 of (2x + 1) dx.

Calculate ∫ from 1 to 3 of (2x + 1) dx.

Correct Answer: 10

Step 1: Identify the function to integrate, which is (2x + 1).
Step 2: Find the antiderivative of (2x + 1). The antiderivative is x^2 + x.
Step 3: Set up the definite integral from 1 to 3 using the antiderivative: [x^2 + x] from 1 to 3.
Step 4: Calculate the value of the antiderivative at the upper limit (3): (3^2 + 3) = (9 + 3) = 12.
Step 5: Calculate the value of the antiderivative at the lower limit (1): (1^2 + 1) = (1 + 1) = 2.
Step 6: Subtract the lower limit result from the upper limit result: 12 - 2 = 10.

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.