Integral von $$$x^{8} - x^{2}$$$

Bestimme $$$\int \left(x^{8} - x^{2}\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(x^{8} - x^{2}\right)d x}}} = {\color{red}{\left(- \int{x^{2} d x} + \int{x^{8} d x}\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=8$$$ an:

$$- \int{x^{2} d x} + {\color{red}{\int{x^{8} d x}}}=- \int{x^{2} d x} + {\color{red}{\frac{x^{1 + 8}}{1 + 8}}}=- \int{x^{2} d x} + {\color{red}{\left(\frac{x^{9}}{9}\right)}}$$

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

$$\frac{x^{9}}{9} - {\color{red}{\int{x^{2} d x}}}=\frac{x^{9}}{9} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{9}}{9} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Daher,

$$\int{\left(x^{8} - x^{2}\right)d x} = \frac{x^{9}}{9} - \frac{x^{3}}{3}$$

Vereinfachen:

$$\int{\left(x^{8} - x^{2}\right)d x} = \frac{x^{3} \left(x^{6} - 3\right)}{9}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(x^{8} - x^{2}\right)d x} = \frac{x^{3} \left(x^{6} - 3\right)}{9}+C$$

$$$\int \left(x^{8} - x^{2}\right)\, dx = \frac{x^{3} \left(x^{6} - 3\right)}{9} + C$$$A