Evaluate the integral ∫ from 1 to 3 of (2x + 1) dx.

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Question: Evaluate the integral ∫ from 1 to 3 of (2x + 1) dx.

Options:

Correct Answer: 8

Solution:

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

Evaluate the integral ∫ from 1 to 3 of (2x + 1) dx.

Evaluate the integral ∫ from 1 to 3 of (2x + 1) dx.

Step 1: Identify the integral you need to evaluate: ∫ from 1 to 3 of (2x + 1) dx.
Step 2: Find the antiderivative of the function (2x + 1). The antiderivative is x^2 + x.
Step 3: Write down the antiderivative with limits: [x^2 + x] from 1 to 3.
Step 4: Substitute the upper limit (3) into the antiderivative: (3^2 + 3) = (9 + 3) = 12.
Step 5: Substitute the lower limit (1) into the antiderivative: (1^2 + 1) = (1 + 1) = 2.
Step 6: Subtract the result of the lower limit from the upper limit: 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.