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

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

Step 1: Identify the function to integrate, which is (x^2 + 4x + 4).
Step 2: Find the antiderivative of the function. The antiderivative of x^2 is x^3/3, the antiderivative of 4x is 2x^2, and the antiderivative of 4 is 4x.
Step 3: Combine the antiderivatives to get the complete antiderivative: (x^3/3 + 2x^2 + 4x).
Step 4: Evaluate the antiderivative from the lower limit (0) to the upper limit (1).
Step 5: Substitute 1 into the antiderivative: (1^3/3 + 2*1^2 + 4*1) = (1/3 + 2 + 4).
Step 6: Simplify the expression: 1/3 + 2 + 4 = 1/3 + 6 = 19/3.
Step 7: Substitute 0 into the antiderivative: (0^3/3 + 2*0^2 + 4*0) = 0.
Step 8: Subtract the value at the lower limit from the value at the upper limit: (19/3 - 0) = 19/3.

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