Evaluate the integral ∫ (3x^2 + 2x + 1) dx.

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

Options:

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

Solution:

The integral of 3x^2 is x^3, the integral of 2x is x^2, and the integral of 1 is x. Therefore, ∫ (3x^2 + 2x + 1) dx = x^3 + x^2 + x + C.

Evaluate the integral ∫ (3x^2 + 2x + 1) dx.

Evaluate the integral ∫ (3x^2 + 2x + 1) dx.

Step 1: Identify the function to integrate, which is (3x^2 + 2x + 1).
Step 2: Break down the integral into parts: ∫(3x^2) dx, ∫(2x) dx, and ∫(1) dx.
Step 3: Find the integral of 3x^2. The integral is x^3 (because you add 1 to the exponent and divide by the new exponent).
Step 4: Find the integral of 2x. The integral is x^2 (again, add 1 to the exponent and divide by the new exponent).
Step 5: Find the integral of 1. The integral is x (since the integral of a constant is the constant times x).
Step 6: Combine all the results from Steps 3, 4, and 5. You get x^3 + x^2 + x.
Step 7: Add the constant of integration, C, to the final result. So, the final answer is x^3 + x^2 + x + C.

Integration of Polynomials – The question tests the ability to integrate polynomial functions using the power rule.