Evaluate ∫ from 0 to 1 of (x^3 + 3x^2 + 3x + 1) dx.

Evaluate ∫ from 0 to 1 of (x^3 + 3x^2 + 3x + 1) dx.

Correct Answer: 4

Step 1: Identify the integral you need to evaluate: ∫ from 0 to 1 of (x^3 + 3x^2 + 3x + 1) dx.
Step 2: Find the antiderivative of the function (x^3 + 3x^2 + 3x + 1).
Step 3: The antiderivative is calculated as follows: For x^3, the antiderivative is (x^4)/4; for 3x^2, it is 3(x^3)/3 = x^3; for 3x, it is (3/2)x^2; and for 1, it is x.
Step 4: Combine the antiderivatives: (x^4)/4 + x^3 + (3/2)x^2 + x.
Step 5: Now, evaluate this antiderivative from 0 to 1: Substitute 1 into the antiderivative: (1^4)/4 + (1^3) + (3/2)(1^2) + (1) = 1/4 + 1 + 3/2 + 1.
Step 6: Calculate the result: 1/4 + 1 + 3/2 + 1 = 1/4 + 1 + 1.5 + 1 = 1/4 + 3.5 = 1/4 + 14/4 = 15/4 = 4.

Definite Integrals – The question tests the ability to evaluate a definite integral, which involves finding the area under the curve of a polynomial function between specified limits.
Polynomial Integration – The question requires knowledge of how to integrate polynomial functions term by term.
Fundamental Theorem of Calculus – The question assesses understanding of applying the Fundamental Theorem of Calculus to evaluate the integral at the upper and lower limits.