Evaluate the integral ∫(2 to 3) (x^3 - 3x^2 + 2) dx. (2023)

Evaluate the integral ∫(2 to 3) (x^3 - 3x^2 + 2) dx. (2023)

Step 1: Identify the integral you need to evaluate: ∫(2 to 3) (x^3 - 3x^2 + 2) dx.
Step 2: Find the antiderivative of the function x^3 - 3x^2 + 2. This means you need to integrate each term separately.
Step 3: The antiderivative of x^3 is (x^4)/4, the antiderivative of -3x^2 is -x^3, and the antiderivative of 2 is 2x.
Step 4: Combine these results to get the complete antiderivative: (x^4)/4 - x^3 + 2x.
Step 5: Now, evaluate this antiderivative from 2 to 3. This means you will calculate it at x = 3 and at x = 2.
Step 6: Calculate the value at x = 3: (3^4)/4 - (3^3) + 2(3) = (81/4) - 27 + 6.
Step 7: Calculate the value at x = 2: (2^4)/4 - (2^3) + 2(2) = (16/4) - 8 + 4.
Step 8: Subtract the value at x = 2 from the value at x = 3 to find the definite integral: [(81/4 - 27 + 6) - (16/4 - 8 + 4)].
Step 9: Simplify the expression to find the final answer.

Definite Integral Evaluation – The question tests the ability to evaluate a definite integral using the Fundamental Theorem of Calculus.
Polynomial Integration – The integral involves integrating a polynomial function, which requires applying power rules for integration.
Substitution of Limits – The question assesses the understanding of correctly substituting the upper and lower limits after finding the antiderivative.