Calculate ∫ from 0 to 1 of (x^4 - 2x^2 + 1) dx.

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Question: Calculate ∫ from 0 to 1 of (x^4 - 2x^2 + 1) dx.

Options:

Correct Answer: 2/3

Solution:

The integral evaluates to [x^5/5 - 2x^3/3 + x] from 0 to 1 = (1/5 - 2/3 + 1) = (15/15 - 10/15 + 3/15) = 8/15.

Calculate ∫ from 0 to 1 of (x^4 - 2x^2 + 1) dx.

Calculate ∫ from 0 to 1 of (x^4 - 2x^2 + 1) dx.

Step 1: Identify the function to integrate, which is (x^4 - 2x^2 + 1).
Step 2: Find the antiderivative of the function. This means we need to integrate each term separately.
Step 3: The antiderivative of x^4 is (x^5)/5.
Step 4: The antiderivative of -2x^2 is -2(x^3)/3.
Step 5: The antiderivative of 1 is x.
Step 6: Combine the antiderivatives: (x^5)/5 - (2x^3)/3 + x.
Step 7: Write the complete antiderivative: (x^5)/5 - (2x^3)/3 + x.
Step 8: Evaluate the antiderivative from 0 to 1. This means we will calculate the value at 1 and subtract the value at 0.
Step 9: Calculate the value at x = 1: (1^5)/5 - (2(1^3))/3 + 1 = 1/5 - 2/3 + 1.
Step 10: Calculate the value at x = 0: (0^5)/5 - (2(0^3))/3 + 0 = 0.
Step 11: Subtract the value at 0 from the value at 1: (1/5 - 2/3 + 1) - 0 = 1/5 - 2/3 + 1.
Step 12: Convert all terms to have a common denominator of 15: 1/5 = 3/15, -2/3 = -10/15, and 1 = 15/15.
Step 13: Combine the fractions: (3/15 - 10/15 + 15/15) = (3 - 10 + 15)/15 = 8/15.

Definite Integral Calculation – The question tests the ability to compute a definite integral of a polynomial function over a specified interval.
Polynomial Integration – It assesses knowledge of integrating polynomial functions using the power rule.
Evaluation of Limits – The question requires evaluating the antiderivative at the upper and lower limits of integration.