Integral of $$$x^{2} - \frac{1}{x^{22}}$$$

Find $$$\int \left(x^{2} - \frac{1}{x^{22}}\right)\, dx$$$.

Integrate term by term:

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

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

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

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

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

Therefore,

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

Simplify:

$$\int{\left(x^{2} - \frac{1}{x^{22}}\right)d x} = \frac{7 x^{24} + 1}{21 x^{21}}$$

Add the constant of integration:

$$\int{\left(x^{2} - \frac{1}{x^{22}}\right)d x} = \frac{7 x^{24} + 1}{21 x^{21}}+C$$

$$$\int \left(x^{2} - \frac{1}{x^{22}}\right)\, dx = \frac{7 x^{24} + 1}{21 x^{21}} + C$$$A