Integral of $$$- x^{5} \left(2 x - 6\right)$$$

Find $$$\int \left(- x^{5} \left(2 x - 6\right)\right)\, dx$$$.

The input is rewritten: $$$\int{\left(- x^{5} \left(2 x - 6\right)\right)d x}=\int{x^{5} \left(6 - 2 x\right) d x}$$$.

Simplify the integrand:

$${\color{red}{\int{x^{5} \left(6 - 2 x\right) d x}}} = {\color{red}{\int{2 x^{5} \left(3 - x\right) d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = x^{5} \left(3 - x\right)$$$:

$${\color{red}{\int{2 x^{5} \left(3 - x\right) d x}}} = {\color{red}{\left(2 \int{x^{5} \left(3 - x\right) d x}\right)}}$$

Expand the expression:

$$2 {\color{red}{\int{x^{5} \left(3 - x\right) d x}}} = 2 {\color{red}{\int{\left(- x^{6} + 3 x^{5}\right)d x}}}$$

Integrate term by term:

$$2 {\color{red}{\int{\left(- x^{6} + 3 x^{5}\right)d x}}} = 2 {\color{red}{\left(\int{3 x^{5} d x} - \int{x^{6} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=6$$$:

$$2 \int{3 x^{5} d x} - 2 {\color{red}{\int{x^{6} d x}}}=2 \int{3 x^{5} d x} - 2 {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}=2 \int{3 x^{5} d x} - 2 {\color{red}{\left(\frac{x^{7}}{7}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = x^{5}$$$:

$$- \frac{2 x^{7}}{7} + 2 {\color{red}{\int{3 x^{5} d x}}} = - \frac{2 x^{7}}{7} + 2 {\color{red}{\left(3 \int{x^{5} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=5$$$:

$$- \frac{2 x^{7}}{7} + 6 {\color{red}{\int{x^{5} d x}}}=- \frac{2 x^{7}}{7} + 6 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=- \frac{2 x^{7}}{7} + 6 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Therefore,

$$\int{x^{5} \left(6 - 2 x\right) d x} = - \frac{2 x^{7}}{7} + x^{6}$$

Simplify:

$$\int{x^{5} \left(6 - 2 x\right) d x} = \frac{x^{6} \left(7 - 2 x\right)}{7}$$

Add the constant of integration:

$$\int{x^{5} \left(6 - 2 x\right) d x} = \frac{x^{6} \left(7 - 2 x\right)}{7}+C$$

$$$\int \left(- x^{5} \left(2 x - 6\right)\right)\, dx = \frac{x^{6} \left(7 - 2 x\right)}{7} + C$$$A