Step 1: Identify the integral to evaluate: ∫_0^1 (x^3 + 2x^2) dx.
Step 2: Break down the integral into two parts: ∫_0^1 x^3 dx + ∫_0^1 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).
Step 8: Find a common denominator to add 1/4 and 2/3, which 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 question tests the ability to evaluate a definite integral by finding the antiderivative and applying the Fundamental Theorem of Calculus.
Polynomial Integration – The question involves integrating a polynomial function, which requires knowledge of basic integration rules for powers of x.
