Evaluate ∫(6x^2 + 3)dx. (2022)

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Question: Evaluate ∫(6x^2 + 3)dx. (2022)

Options:

Correct Answer: 2x^3 + 3x + C

Exam Year: 2022

Solution:

Integrating term by term: ∫6x^2dx = 2x^3 and ∫3dx = 3x. Thus, ∫(6x^2 + 3)dx = 2x^3 + 3x + C.

Evaluate ∫(6x^2 + 3)dx. (2022)

Evaluate ∫(6x^2 + 3)dx. (2022)

Step 1: Identify the integral you need to evaluate: ∫(6x^2 + 3)dx.
Step 2: Break the integral into two separate parts: ∫6x^2dx + ∫3dx.
Step 3: Evaluate the first part: ∫6x^2dx. To do this, increase the exponent of x by 1 (from 2 to 3) and divide by the new exponent: (6/3)x^3 = 2x^3.
Step 4: Evaluate the second part: ∫3dx. The integral of a constant (3) is simply 3 times x: 3x.
Step 5: Combine the results from Step 3 and Step 4: 2x^3 + 3x.
Step 6: Don't forget to add the constant of integration (C) at the end: 2x^3 + 3x + C.

Integration – The process of finding the integral of a function, which involves determining the antiderivative.
Polynomial Functions – Understanding how to integrate polynomial terms individually.
Constant of Integration – Recognizing the importance of adding a constant (C) to the result of an indefinite integral.