Calculate ∫ from 0 to 1 of (4x^3 - 2x^2 + x) dx.

Calculate ∫ from 0 to 1 of (4x^3 - 2x^2 + x) dx.

Step 1: Identify the function to integrate, which is f(x) = 4x^3 - 2x^2 + x.
Step 2: Find the antiderivative (indefinite integral) of f(x). This means we need to integrate each term separately.
Step 3: Integrate 4x^3. The antiderivative is (4/4)x^4 = x^4.
Step 4: Integrate -2x^2. The antiderivative is (-2/3)x^3.
Step 5: Integrate x. The antiderivative is (1/2)x^2.
Step 6: Combine the antiderivatives. The complete antiderivative is x^4 - (2/3)x^3 + (1/2)x^2.
Step 7: Evaluate the antiderivative from 0 to 1. This means we will calculate the value at x = 1 and subtract the value at x = 0.
Step 8: Calculate the value at x = 1: 1^4 - (2/3)(1^3) + (1/2)(1^2) = 1 - (2/3) + (1/2).
Step 9: Simplify the expression: 1 - (2/3) + (1/2) = (6/6) - (4/6) + (3/6) = (6 - 4 + 3)/6 = 5/6.
Step 10: Calculate the value at x = 0: 0^4 - (2/3)(0^3) + (1/2)(0^2) = 0.
Step 11: Subtract the value at x = 0 from the value at x = 1: (5/6) - 0 = 5/6.
Step 12: The final answer is 5/6.

Definite Integral – The process of calculating the area under a curve defined by a polynomial function over a specified interval.
Polynomial Integration – Applying the power rule for integration to find the antiderivative of polynomial terms.
Fundamental Theorem of Calculus – Connecting differentiation and integration, allowing evaluation of definite integrals using antiderivatives.