Integral of $$$- 3 x^{21} \left(x - 4\right)$$$

Find $$$\int \left(- 3 x^{21} \left(x - 4\right)\right)\, dx$$$.

The input is rewritten: $$$\int{\left(- 3 x^{21} \left(x - 4\right)\right)d x}=\int{x^{21} \left(12 - 3 x\right) d x}$$$.

Simplify the integrand:

$${\color{red}{\int{x^{21} \left(12 - 3 x\right) d x}}} = {\color{red}{\int{3 x^{21} \left(4 - x\right) d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = x^{21} \left(4 - x\right)$$$:

$${\color{red}{\int{3 x^{21} \left(4 - x\right) d x}}} = {\color{red}{\left(3 \int{x^{21} \left(4 - x\right) d x}\right)}}$$

Expand the expression:

$$3 {\color{red}{\int{x^{21} \left(4 - x\right) d x}}} = 3 {\color{red}{\int{\left(- x^{22} + 4 x^{21}\right)d x}}}$$

Integrate term by term:

$$3 {\color{red}{\int{\left(- x^{22} + 4 x^{21}\right)d x}}} = 3 {\color{red}{\left(\int{4 x^{21} d x} - \int{x^{22} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=22$$$:

$$3 \int{4 x^{21} d x} - 3 {\color{red}{\int{x^{22} d x}}}=3 \int{4 x^{21} d x} - 3 {\color{red}{\frac{x^{1 + 22}}{1 + 22}}}=3 \int{4 x^{21} d x} - 3 {\color{red}{\left(\frac{x^{23}}{23}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = x^{21}$$$:

$$- \frac{3 x^{23}}{23} + 3 {\color{red}{\int{4 x^{21} d x}}} = - \frac{3 x^{23}}{23} + 3 {\color{red}{\left(4 \int{x^{21} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=21$$$:

$$- \frac{3 x^{23}}{23} + 12 {\color{red}{\int{x^{21} d x}}}=- \frac{3 x^{23}}{23} + 12 {\color{red}{\frac{x^{1 + 21}}{1 + 21}}}=- \frac{3 x^{23}}{23} + 12 {\color{red}{\left(\frac{x^{22}}{22}\right)}}$$

Therefore,

$$\int{x^{21} \left(12 - 3 x\right) d x} = - \frac{3 x^{23}}{23} + \frac{6 x^{22}}{11}$$

Simplify:

$$\int{x^{21} \left(12 - 3 x\right) d x} = \frac{3 x^{22} \left(46 - 11 x\right)}{253}$$

Add the constant of integration:

$$\int{x^{21} \left(12 - 3 x\right) d x} = \frac{3 x^{22} \left(46 - 11 x\right)}{253}+C$$

$$$\int \left(- 3 x^{21} \left(x - 4\right)\right)\, dx = \frac{3 x^{22} \left(46 - 11 x\right)}{253} + C$$$A