Find the value of the integral ∫(0 to 1) (x^2 + 2x)dx.

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

Options:

Correct Answer: 2

Solution:

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

Find the value of the integral ∫(0 to 1) (x^2 + 2x)dx.

Find the value of the integral ∫(0 to 1) (x^2 + 2x)dx.

Correct Answer: 4/3

Step 1: Identify the integral you need to solve: ∫(0 to 1) (x^2 + 2x)dx.
Step 2: Break down the integral into two parts: ∫(0 to 1) x^2 dx + ∫(0 to 1) 2x dx.
Step 3: Find the antiderivative of x^2. The antiderivative is (1/3)x^3.
Step 4: Find the antiderivative of 2x. The antiderivative is x^2.
Step 5: Combine the antiderivatives: (1/3)x^3 + x^2.
Step 6: Evaluate the combined antiderivative from 0 to 1: [(1/3)(1)^3 + (1)^2] - [(1/3)(0)^3 + (0)^2].
Step 7: Calculate the value at the upper limit (1): (1/3) + 1 = 4/3.
Step 8: Calculate the value at the lower limit (0): 0.
Step 9: Subtract the lower limit value from the upper limit value: (4/3) - 0 = 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.