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

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 ):

[