Evaluate ∫ (3x^2 - 2x + 1) dx. What is the answer?

Evaluate ∫ (3x^2 - 2x + 1) dx. What is the answer?

• A. x^3 - x^2 + x + C
• B. x^3 - 2x^2 + x + C
• C. x^3 + x^2 + x + C
• D. 3x^3 - x^2 + C

Answer: (A)

To evaluate the integral ∫ (3x² - 2x + 1) dx, we will integrate each term of the polynomial separately. 1. For the first term, 3x²: The integral of x² is (1/3)x³. Therefore, the integral of 3x² is 3 * (1/3)x³ = x³. 2. For the second term, -2x: The integral of x is (1/2)x². Therefore, the integral of -2x is -2 * (1/2)x² = -x². 3. For the third term, +1: The integral of 1 with respect to x is simply x. Combining these results, we have: ∫ (3x² - 2x + 1) dx = x³ - x² + x + C, where C is the constant of integration. This matches with the first choice provided, confirming that it is indeed the correct evaluation of the integral. The integral calculus rules applied here—specifically the power rule—accurately yield the sum of these integrated terms, leading us directly to the solution.

To evaluate the integral ∫ (3x² - 2x + 1) dx, we will integrate each term of the polynomial separately.

1. For the first term, 3x²:

The integral of x² is (1/3)x³. Therefore, the integral of 3x² is 3 * (1/3)x³ = x³.

2. For the second term, -2x:

The integral of x is (1/2)x². Therefore, the integral of -2x is -2 * (1/2)x² = -x².

3. For the third term, +1:

The integral of 1 with respect to x is simply x.

Combining these results, we have:

∫ (3x² - 2x + 1) dx = x³ - x² + x + C, where C is the constant of integration.

This matches with the first choice provided, confirming that it is indeed the correct evaluation of the integral. The integral calculus rules applied here—specifically the power rule—accurately yield the sum of these integrated terms, leading us directly to the solution.