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

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

Options:

Correct Answer: 0

Solution:

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

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

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

Correct Answer: 1/3

Step 1: Identify the function to integrate, which is (x^2 - 2x + 1).
Step 2: Find the antiderivative of the function. The antiderivative of x^2 is x^3/3, the antiderivative of -2x is -x^2, and the antiderivative of 1 is x.
Step 3: Combine the antiderivatives to get the complete antiderivative: (x^3/3 - x^2 + x).
Step 4: Evaluate the antiderivative from the lower limit (0) to the upper limit (1).
Step 5: Substitute the upper limit (1) into the antiderivative: (1^3/3 - 1^2 + 1) = (1/3 - 1 + 1).
Step 6: Simplify the expression: 1/3 - 1 + 1 = 1/3.

Definite Integral – The question tests the ability to evaluate a definite integral of a polynomial function over a specified interval.
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