What is the integral of the polynomial ∫ (5x^3 + 2x^2 - x + 1) dx?

What is the integral of the polynomial ∫ (5x^3 + 2x^2 - x + 1) dx?

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

Answer: (A)

To find the integral of the polynomial \( \int (5x^3 + 2x^2 - x + 1) \, dx \), we integrate each term individually. 1. For the term \( 5x^3 \), applying the power rule for integration, we add 1 to the exponent (3 + 1 = 4) and divide by the new exponent: \[ \int 5x^3 \, dx = 5 \cdot \frac{x^4}{4} = \frac{5}{4}x^4. \] 2. For \( 2x^2 \), we again apply the power rule: \[ \int 2x^2 \, dx = 2 \cdot \frac{x^3}{3} = \frac{2}{3}x^3. \] 3. Next, for \( -x \): \[ \int -x \, dx = -\frac{x^2}{2} = -\frac{1}{2}x^2. \] 4. Finally, for the constant term \( 1 \): \[

To find the integral of the polynomial ( \int (5x^3 + 2x^2 - x + 1) , dx ), we integrate each term individually.

1. For the term ( 5x^3 ), applying the power rule for integration, we add 1 to the exponent (3 + 1 = 4) and divide by the new exponent:

[

\int 5x^3 , dx = 5 \cdot \frac{x^4}{4} = \frac{5}{4}x^4.

]

2. For ( 2x^2 ), we again apply the power rule:

[

\int 2x^2 , dx = 2 \cdot \frac{x^3}{3} = \frac{2}{3}x^3.

]

3. Next, for ( -x ):

[

\int -x , dx = -\frac{x^2}{2} = -\frac{1}{2}x^2.

]

4. Finally, for the constant term ( 1 ):

[