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

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

• A. (1/2)x^4 - x^3 + (1/2)x^2 + C
• B. (2/5)x^5 - x^3 + C
• C. (1/3)x^3 - (3/2)x^2 + C
• D. (1/4)x^4 - (3/3)x^2 + C

Answer: (A)

To evaluate the integral ∫ (2x^3 - 3x^2 + x) dx, we apply the power rule of integration, which states that the integral of x^n is (1/(n+1))x^(n+1) + C, for any real number n not equal to -1. In this case, we break down the integral into the sum of the integrals of each term: 1. For the first term, 2x^3: ∫ 2x^3 dx = 2 * (1/(3 + 1))x^(3 + 1) = 2 * (1/4)x^4 = (1/2)x^4. 2. For the second term, -3x^2: ∫ -3x^2 dx = -3 * (1/(2 + 1))x^(2 + 1) = -3 * (1/3)x^3 = -x^3. 3. For the third term, x: ∫ x dx = (1/(1 + 1))x^(1 + 1) = (1/2)x^2. Putting it all together

To evaluate the integral ∫ (2x^3 - 3x^2 + x) dx, we apply the power rule of integration, which states that the integral of x^n is (1/(n+1))x^(n+1) + C, for any real number n not equal to -1.

In this case, we break down the integral into the sum of the integrals of each term:

1. For the first term, 2x^3:

∫ 2x^3 dx = 2 * (1/(3 + 1))x^(3 + 1) = 2 * (1/4)x^4 = (1/2)x^4.

2. For the second term, -3x^2:

∫ -3x^2 dx = -3 * (1/(2 + 1))x^(2 + 1) = -3 * (1/3)x^3 = -x^3.

3. For the third term, x:

∫ x dx = (1/(1 + 1))x^(1 + 1) = (1/2)x^2.

Putting it all together