Integral of $$$x^{2} \left(2 x - 2\right) - 81 x$$$

Find $$$\int \left(x^{2} \left(2 x - 2\right) - 81 x\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(x^{2} \left(2 x - 2\right) - 81 x\right)d x}}} = {\color{red}{\left(- \int{81 x d x} + \int{x^{2} \left(2 x - 2\right) d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=81$$$ and $$$f{\left(x \right)} = x$$$:

$$\int{x^{2} \left(2 x - 2\right) d x} - {\color{red}{\int{81 x d x}}} = \int{x^{2} \left(2 x - 2\right) d x} - {\color{red}{\left(81 \int{x d x}\right)}}$$

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

$$\int{x^{2} \left(2 x - 2\right) d x} - 81 {\color{red}{\int{x d x}}}=\int{x^{2} \left(2 x - 2\right) d x} - 81 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\int{x^{2} \left(2 x - 2\right) d x} - 81 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Simplify the integrand:

$$- \frac{81 x^{2}}{2} + {\color{red}{\int{x^{2} \left(2 x - 2\right) d x}}} = - \frac{81 x^{2}}{2} + {\color{red}{\int{2 x^{2} \left(x - 1\right) d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = x^{2} \left(x - 1\right)$$$:

$$- \frac{81 x^{2}}{2} + {\color{red}{\int{2 x^{2} \left(x - 1\right) d x}}} = - \frac{81 x^{2}}{2} + {\color{red}{\left(2 \int{x^{2} \left(x - 1\right) d x}\right)}}$$

Expand the expression:

$$- \frac{81 x^{2}}{2} + 2 {\color{red}{\int{x^{2} \left(x - 1\right) d x}}} = - \frac{81 x^{2}}{2} + 2 {\color{red}{\int{\left(x^{3} - x^{2}\right)d x}}}$$

Integrate term by term:

$$- \frac{81 x^{2}}{2} + 2 {\color{red}{\int{\left(x^{3} - x^{2}\right)d x}}} = - \frac{81 x^{2}}{2} + 2 {\color{red}{\left(- \int{x^{2} d x} + \int{x^{3} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=3$$$:

$$- \frac{81 x^{2}}{2} - 2 \int{x^{2} d x} + 2 {\color{red}{\int{x^{3} d x}}}=- \frac{81 x^{2}}{2} - 2 \int{x^{2} d x} + 2 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \frac{81 x^{2}}{2} - 2 \int{x^{2} d x} + 2 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

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

$$\frac{x^{4}}{2} - \frac{81 x^{2}}{2} - 2 {\color{red}{\int{x^{2} d x}}}=\frac{x^{4}}{2} - \frac{81 x^{2}}{2} - 2 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{4}}{2} - \frac{81 x^{2}}{2} - 2 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(x^{2} \left(2 x - 2\right) - 81 x\right)d x} = \frac{x^{4}}{2} - \frac{2 x^{3}}{3} - \frac{81 x^{2}}{2}$$

Simplify:

$$\int{\left(x^{2} \left(2 x - 2\right) - 81 x\right)d x} = \frac{x^{2} \left(3 x^{2} - 4 x - 243\right)}{6}$$

Add the constant of integration:

$$\int{\left(x^{2} \left(2 x - 2\right) - 81 x\right)d x} = \frac{x^{2} \left(3 x^{2} - 4 x - 243\right)}{6}+C$$

$$$\int \left(x^{2} \left(2 x - 2\right) - 81 x\right)\, dx = \frac{x^{2} \left(3 x^{2} - 4 x - 243\right)}{6} + C$$$A