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

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

Options:

Correct Answer: 8

Solution:

The integral ∫(x^2 + 2x)dx = [(1/3)x^3 + x^2] from 1 to 2 = 8.

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

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

Step 1: Identify the integral you need to evaluate: ∫(1 to 2) (x^2 + 2x)dx.
Step 2: Find the antiderivative of the function (x^2 + 2x). The antiderivative is (1/3)x^3 + x^2.
Step 3: Write down the antiderivative: F(x) = (1/3)x^3 + x^2.
Step 4: Evaluate the antiderivative at the upper limit (x = 2): F(2) = (1/3)(2^3) + (2^2).
Step 5: Calculate F(2): F(2) = (1/3)(8) + 4 = (8/3) + 4 = (8/3) + (12/3) = 20/3.
Step 6: Evaluate the antiderivative at the lower limit (x = 1): F(1) = (1/3)(1^3) + (1^2).
Step 7: Calculate F(1): F(1) = (1/3)(1) + 1 = (1/3) + 1 = (1/3) + (3/3) = 4/3.
Step 8: Subtract the value at the lower limit from the value at the upper limit: F(2) - F(1) = (20/3) - (4/3).
Step 9: Calculate the result: (20/3) - (4/3) = (20 - 4)/3 = 16/3.
Step 10: The final answer for the integral ∫(1 to 2) (x^2 + 2x)dx is 16/3.

Definite Integral – The process of calculating the area under a curve defined by a function over a specific interval.
Fundamental Theorem of Calculus – This theorem connects differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.
Polynomial Integration – The technique of integrating polynomial functions by applying the power rule.