Integral of $$$3 x - \frac{1}{x^{22}}$$$

Find $$$\int \left(3 x - \frac{1}{x^{22}}\right)\, dx$$$.

Integrate term by term:

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

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

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

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

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

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

$$3 {\color{red}{\int{x d x}}} + \frac{1}{21 x^{21}}=3 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}} + \frac{1}{21 x^{21}}=3 {\color{red}{\left(\frac{x^{2}}{2}\right)}} + \frac{1}{21 x^{21}}$$

Therefore,

$$\int{\left(3 x - \frac{1}{x^{22}}\right)d x} = \frac{3 x^{2}}{2} + \frac{1}{21 x^{21}}$$

Simplify:

$$\int{\left(3 x - \frac{1}{x^{22}}\right)d x} = \frac{63 x^{23} + 2}{42 x^{21}}$$

Add the constant of integration:

$$\int{\left(3 x - \frac{1}{x^{22}}\right)d x} = \frac{63 x^{23} + 2}{42 x^{21}}+C$$

$$$\int \left(3 x - \frac{1}{x^{22}}\right)\, dx = \frac{63 x^{23} + 2}{42 x^{21}} + C$$$A