Find ∫ (6x^2 - 4) dx. (2019)

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Question: Find ∫ (6x^2 - 4) dx. (2019)

Options:

Correct Answer: 2x^3 - 4x + C

Solution:

The integral is (6/3)x^3 - 4x + C = 2x^3 - 4x + C.

Find ∫ (6x^2 - 4) dx. (2019)

Find ∫ (6x^2 - 4) dx. (2019)

Step 1: Identify the function to integrate, which is 6x^2 - 4.
Step 2: Apply the power rule for integration. The power rule states that ∫x^n dx = (1/(n+1))x^(n+1) + C, where n is a constant.
Step 3: For the term 6x^2, apply the power rule: ∫6x^2 dx = 6 * (1/(2+1))x^(2+1) = 6 * (1/3)x^3 = 2x^3.
Step 4: For the constant term -4, the integral is simply -4x, since ∫k dx = kx, where k is a constant.
Step 5: Combine the results from Step 3 and Step 4: 2x^3 - 4x.
Step 6: Add the constant of integration, C, to the final result: 2x^3 - 4x + C.

Integration of Polynomials – The question tests the ability to integrate polynomial functions using the power rule.
Constant of Integration – It assesses the understanding of including the constant of integration (C) in indefinite integrals.