Find the value of ∫ from 0 to 1 of (x^4 - 4x^3 + 6x^2 - 4x + 1) dx.

Find the value of ∫ from 0 to 1 of (x^4 - 4x^3 + 6x^2 - 4x + 1) dx.

Correct Answer: 0

Step 1: Identify the function to integrate, which is f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1.
Step 2: Find the antiderivative (indefinite integral) of f(x). This means we need to integrate each term separately.
Step 3: Integrate each term: The integral of x^4 is (x^5)/5, the integral of -4x^3 is -x^4, the integral of 6x^2 is 2x^3, the integral of -4x is -2x^2, and the integral of 1 is x.
Step 4: Combine the results of the integrations: The antiderivative is (x^5)/5 - x^4 + 2x^3 - 2x^2 + x.
Step 5: Evaluate the definite integral from 0 to 1. This means we will calculate the antiderivative at 1 and subtract the value of the antiderivative at 0.
Step 6: Calculate the antiderivative at x = 1: (1^5)/5 - (1^4) + 2(1^3) - 2(1^2) + 1 = 1/5 - 1 + 2 - 2 + 1.
Step 7: Simplify the expression: 1/5 - 1 + 2 - 2 + 1 = 1/5 + 0 = 1/5.
Step 8: Calculate the antiderivative at x = 0: (0^5)/5 - (0^4) + 2(0^3) - 2(0^2) + 0 = 0.
Step 9: Subtract the value at x = 0 from the value at x = 1: (1/5) - (0) = 1/5.
Step 10: The final result of the definite integral is 1/5.

Definite Integral – The question tests the ability to evaluate a definite integral of a polynomial function over a specified interval.
Polynomial Integration – It assesses the understanding of integrating polynomial functions and applying the Fundamental Theorem of Calculus.
Evaluation of Limits – The question requires evaluating the antiderivative at the upper and lower limits of integration.