Evaluate the integral ∫ from 0 to 1 of (x^2 + 2x) dx.

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Question: Evaluate the integral ∫ from 0 to 1 of (x^2 + 2x) dx.

Options:

Correct Answer: 2

Solution:

The integral evaluates to [x^3/3 + x^2] from 0 to 1 = (1/3 + 1) - (0) = 4/3.

Evaluate the integral ∫ from 0 to 1 of (x^2 + 2x) dx.

Evaluate the integral ∫ from 0 to 1 of (x^2 + 2x) dx.

Correct Answer: 4/3

Step 1: Identify the integral you need to evaluate: ∫ from 0 to 1 of (x^2 + 2x) dx.
Step 2: Find the antiderivative of the function (x^2 + 2x). The antiderivative is (x^3/3 + x^2).
Step 3: Evaluate the antiderivative at the upper limit (1): (1^3/3 + 1^2) = (1/3 + 1) = 1/3 + 3/3 = 4/3.
Step 4: Evaluate the antiderivative at the lower limit (0): (0^3/3 + 0^2) = 0.
Step 5: Subtract the lower limit evaluation from the upper limit evaluation: (4/3) - (0) = 4/3.

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.
Polynomial Integration – Involves integrating polynomial functions by applying the power rule.