Integral von $$$x^{2} - \frac{1}{x^{21}}$$$

Bestimme $$$\int \left(x^{2} - \frac{1}{x^{21}}\right)\, dx$$$.

Gliedweise integrieren:

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

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=2$$$ an:

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

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=-21$$$ an:

$$\frac{x^{3}}{3} - {\color{red}{\int{\frac{1}{x^{21}} d x}}}=\frac{x^{3}}{3} - {\color{red}{\int{x^{-21} d x}}}=\frac{x^{3}}{3} - {\color{red}{\frac{x^{-21 + 1}}{-21 + 1}}}=\frac{x^{3}}{3} - {\color{red}{\left(- \frac{x^{-20}}{20}\right)}}=\frac{x^{3}}{3} - {\color{red}{\left(- \frac{1}{20 x^{20}}\right)}}$$

Daher,

$$\int{\left(x^{2} - \frac{1}{x^{21}}\right)d x} = \frac{x^{3}}{3} + \frac{1}{20 x^{20}}$$

Vereinfachen:

$$\int{\left(x^{2} - \frac{1}{x^{21}}\right)d x} = \frac{20 x^{23} + 3}{60 x^{20}}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(x^{2} - \frac{1}{x^{21}}\right)d x} = \frac{20 x^{23} + 3}{60 x^{20}}+C$$

$$$\int \left(x^{2} - \frac{1}{x^{21}}\right)\, dx = \frac{20 x^{23} + 3}{60 x^{20}} + C$$$A