Find the value of ∫ from 0 to 1 of (x^4 + 2x^2) dx.

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Question: Find the value of ∫ from 0 to 1 of (x^4 + 2x^2) dx.

Options:

Correct Answer: 1/3

Solution:

The integral evaluates to [x^5/5 + (2/3)x^3] from 0 to 1 = (1/5 + 2/3) = 11/15.

Find the value of ∫ from 0 to 1 of (x^4 + 2x^2) dx.

Find the value of ∫ from 0 to 1 of (x^4 + 2x^2) dx.

Step 1: Identify the integral you need to solve: ∫ from 0 to 1 of (x^4 + 2x^2) dx.
Step 2: Break the integral into two parts: ∫ from 0 to 1 of x^4 dx + ∫ from 0 to 1 of 2x^2 dx.
Step 3: Find the antiderivative of x^4. The antiderivative is (x^5)/5.
Step 4: Find the antiderivative of 2x^2. The antiderivative is (2/3)x^3.
Step 5: Combine the antiderivatives: (x^5)/5 + (2/3)x^3.
Step 6: Evaluate the combined antiderivative from 0 to 1: [(1^5)/5 + (2/3)(1^3)] - [(0^5)/5 + (2/3)(0^3)].
Step 7: Calculate the values: (1/5 + 2/3) - (0 + 0).
Step 8: Find a common denominator to add 1/5 and 2/3. The common denominator is 15.
Step 9: Convert 1/5 to 3/15 and 2/3 to 10/15.
Step 10: Add the fractions: 3/15 + 10/15 = 13/15.

Definite Integral – The question tests the ability to evaluate a definite integral of a polynomial function over a specified interval.
Polynomial Integration – It assesses knowledge of integrating polynomial terms using the power rule.