What is the integral of the polynomial function (3x^2 - x + 7) dx?

To find the integral of the polynomial function ( 3x^2 - x + 7 ), we apply the power rule of integration, which states that the integral of ( x^n ) is ( \frac{x^{n+1}}{n+1} + C ), where ( C ) is the constant of integration.

Let's break down the polynomial:

1. For the term ( 3x^2 ), applying the power rule gives:

[

\int 3x^2 , dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3} = x^3

]

2. For the term ( -x ) (which can be considered as ( -1x^1 )):

[

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

]

3. For the constant term ( 7 ):

[

\int 7 , dx =