Evaluate ∫_0^1 (x^4 - 2x^2 + 1) dx.

Evaluate ∫_0^1 (x^4 - 2x^2 + 1) dx.

Step 1: Identify the integral to evaluate: ∫_0^1 (x^4 - 2x^2 + 1) dx.
Step 2: Break down the expression inside the integral: x^4 - 2x^2 + 1.
Step 3: Find the antiderivative of each term: The antiderivative of x^4 is x^5/5, the antiderivative of -2x^2 is -2/3 x^3, and the antiderivative of 1 is x.
Step 4: Combine the antiderivatives: The complete antiderivative is (x^5/5) - (2/3)x^3 + x.
Step 5: Evaluate the antiderivative from 0 to 1: Substitute 1 into the antiderivative: (1^5/5) - (2/3)(1^3) + (1) = (1/5) - (2/3) + (1).
Step 6: Simplify the expression: Convert to a common denominator (15): (1/5) = (3/15), (-2/3) = (-10/15), and (1) = (15/15).
Step 7: Combine the fractions: (3/15) - (10/15) + (15/15) = (3 - 10 + 15)/15 = 8/15.
Step 8: Final result: The value of the integral is 8/15.

Definite Integral Evaluation – 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.