Evaluate the integral ∫(0 to 1) (x^3 + 2x^2)dx.

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Question: Evaluate the integral ∫(0 to 1) (x^3 + 2x^2)dx.

Options:

Correct Answer: 1/3

Solution:

The integral evaluates to [x^4/4 + 2x^3/3] from 0 to 1 = 1/4 + 2/3 = 11/12.

Evaluate the integral ∫(0 to 1) (x^3 + 2x^2)dx.

Evaluate the integral ∫(0 to 1) (x^3 + 2x^2)dx.

Step 1: Identify the integral to evaluate: ∫(0 to 1) (x^3 + 2x^2)dx.
Step 2: Break the integral into two parts: ∫(0 to 1) x^3 dx + ∫(0 to 1) 2x^2 dx.
Step 3: Find the antiderivative of x^3, which is (x^4)/4.
Step 4: Find the antiderivative of 2x^2, which is (2x^3)/3.
Step 5: Combine the antiderivatives: (x^4)/4 + (2x^3)/3.
Step 6: Evaluate the combined antiderivative from 0 to 1: [(1^4)/4 + (2*1^3)/3] - [(0^4)/4 + (2*0^3)/3].
Step 7: Calculate the values: (1/4 + 2/3) - (0 + 0).
Step 8: Find a common denominator to add 1/4 and 2/3. The common denominator is 12.
Step 9: Convert 1/4 to 3/12 and 2/3 to 8/12.
Step 10: Add the fractions: 3/12 + 8/12 = 11/12.

Definite Integral Evaluation – The process of calculating the area under a curve defined by a function over a specified interval.
Polynomial Integration – Applying the power rule for integration to polynomial functions.