Evaluate the integral ∫_0^1 (x^2 + 2x) dx.

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Question: Evaluate the integral ∫_0^1 (x^2 + 2x) dx.

Options:

Correct Answer: 2

Solution:

∫_0^1 (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 1 = (1/3 + 1) - (0) = 4/3.

Evaluate the integral ∫_0^1 (x^2 + 2x) dx.

Evaluate the integral ∫_0^1 (x^2 + 2x) dx.

Step 1: Identify the integral to evaluate: ∫_0^1 (x^2 + 2x) dx.
Step 2: Break down the integral into two parts: ∫_0^1 x^2 dx + ∫_0^1 2x dx.
Step 3: Find the antiderivative of x^2, which is (x^3)/3.
Step 4: Find the antiderivative of 2x, which is x^2.
Step 5: Combine the antiderivatives: (x^3)/3 + x^2.
Step 6: Evaluate the combined antiderivative from 0 to 1: [(1^3)/3 + (1^2)] - [(0^3)/3 + (0^2)].
Step 7: Calculate the values: (1/3 + 1) - (0) = 1/3 + 1 = 4/3.

Definite Integral – The process of calculating the area under a curve defined by a function over a specific interval.
Polynomial Integration – Applying the power rule for integration to polynomial functions.
Fundamental Theorem of Calculus – Connecting differentiation and integration, allowing evaluation of definite integrals using antiderivatives.