Calculate ∫ from 0 to 1 of (x^4 - 2x^3 + x^2) dx.

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Question: Calculate ∫ from 0 to 1 of (x^4 - 2x^3 + x^2) dx.

Options:

Correct Answer: 1/5

Solution:

The integral evaluates to [x^5/5 - (2/4)x^4 + (1/3)x^3] from 0 to 1 = (1/5 - 1/2 + 1/3) = 1/30.

Calculate ∫ from 0 to 1 of (x^4 - 2x^3 + x^2) dx.

Calculate ∫ from 0 to 1 of (x^4 - 2x^3 + x^2) dx.

Correct Answer: 1/30

Step 1: Identify the function to integrate, which is f(x) = x^4 - 2x^3 + x^2.
Step 2: Find the antiderivative of f(x). This means we need to integrate each term separately.
Step 3: Integrate x^4. The antiderivative is (1/5)x^5.
Step 4: Integrate -2x^3. The antiderivative is -2 * (1/4)x^4 = -1/2 x^4.
Step 5: Integrate x^2. The antiderivative is (1/3)x^3.
Step 6: Combine the antiderivatives. The complete antiderivative is (1/5)x^5 - (1/2)x^4 + (1/3)x^3.
Step 7: Evaluate the antiderivative from 0 to 1. Substitute x = 1 into the antiderivative: (1/5)(1)^5 - (1/2)(1)^4 + (1/3)(1)^3.
Step 8: Calculate the values: (1/5) - (1/2) + (1/3).
Step 9: Find a common denominator to combine the fractions. The common denominator for 5, 2, and 3 is 30.
Step 10: Rewrite each fraction: (1/5) = 6/30, (1/2) = 15/30, (1/3) = 10/30.
Step 11: Combine the fractions: 6/30 - 15/30 + 10/30 = (6 - 15 + 10)/30 = 1/30.
Step 12: The final answer is 1/30.

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.