Integral of $$$2 x^{2} - 2 x$$$

Find $$$\int \left(2 x^{2} - 2 x\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(2 x^{2} - 2 x\right)d x}}} = {\color{red}{\left(- \int{2 x d x} + \int{2 x^{2} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = x$$$:

$$\int{2 x^{2} d x} - {\color{red}{\int{2 x d x}}} = \int{2 x^{2} d x} - {\color{red}{\left(2 \int{x d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=1$$$:

$$\int{2 x^{2} d x} - 2 {\color{red}{\int{x d x}}}=\int{2 x^{2} d x} - 2 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\int{2 x^{2} d x} - 2 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$- x^{2} + {\color{red}{\int{2 x^{2} d x}}} = - x^{2} + {\color{red}{\left(2 \int{x^{2} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$- x^{2} + 2 {\color{red}{\int{x^{2} d x}}}=- x^{2} + 2 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- x^{2} + 2 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(2 x^{2} - 2 x\right)d x} = \frac{2 x^{3}}{3} - x^{2}$$

Simplify:

$$\int{\left(2 x^{2} - 2 x\right)d x} = \frac{x^{2} \left(2 x - 3\right)}{3}$$

Add the constant of integration:

$$\int{\left(2 x^{2} - 2 x\right)d x} = \frac{x^{2} \left(2 x - 3\right)}{3}+C$$

$$$\int \left(2 x^{2} - 2 x\right)\, dx = \frac{x^{2} \left(2 x - 3\right)}{3} + C$$$A