Evaluate ∫(2x^2 + 3x + 1)dx. (2021)

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Question: Evaluate ∫(2x^2 + 3x + 1)dx. (2021)

Options:

Correct Answer: (2/3)x^3 + (3/2)x^2 + x + C

Exam Year: 2021

Solution:

Integrating term by term: ∫2x^2dx = (2/3)x^3, ∫3xdx = (3/2)x^2, and ∫1dx = x. Thus, ∫(2x^2 + 3x + 1)dx = (2/3)x^3 + (3/2)x^2 + x + C.

Evaluate ∫(2x^2 + 3x + 1)dx. (2021)

Evaluate ∫(2x^2 + 3x + 1)dx. (2021)

Step 1: Identify the function to integrate, which is (2x^2 + 3x + 1).
Step 2: Break down the integral into separate parts: ∫(2x^2)dx + ∫(3x)dx + ∫(1)dx.
Step 3: Integrate the first part: ∫(2x^2)dx. The formula for integrating x^n is (1/(n+1))x^(n+1). Here, n=2, so it becomes (2/3)x^3.
Step 4: Integrate the second part: ∫(3x)dx. Using the same formula, n=1, so it becomes (3/2)x^2.
Step 5: Integrate the last part: ∫(1)dx. The integral of 1 is simply x.
Step 6: Combine all the results from the integrations: (2/3)x^3 + (3/2)x^2 + x.
Step 7: Add the constant of integration, C, to the final result.

Integration – The process of finding the integral of a function, which involves calculating the antiderivative.
Polynomial Functions – Understanding how to integrate polynomial expressions term by term.
Constant of Integration – Recognizing the importance of adding the constant of integration (C) after finding the indefinite integral.