How do you calculate ∫ (x^2 - 4x + 4) dx?

To find the integral of the expression ( \int (x^2 - 4x + 4) , dx ), we need to break down the polynomial into its individual terms and integrate each term separately.

1. Integrating ( x^2 ): The antiderivative of ( x^n ) (where ( n ) is any real number) is given by ( \frac{x^{n+1}}{n+1} ). For ( x^2 ):

[

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

]

2. Integrating ( -4x ): Following the same rule, the antiderivative of ( -4x ) is:

[

\int -4x , dx = -4 \cdot \frac{x^{1+1}}{1+1} = -4 \cdot \frac{x^2}{2} = -2x^2.

]

3. Integrating ( 4 ): For the constant term, the integral is:

[