Integral von $$$x^{5} \left(x^{6} - 7\right)^{4}$$$

Bestimme $$$\int x^{5} \left(x^{6} - 7\right)^{4}\, dx$$$.

Sei $$$u=x^{6} - 7$$$.

Dann $$$du=\left(x^{6} - 7\right)^{\prime }dx = 6 x^{5} dx$$$ (die Schritte sind » zu sehen), und es gilt $$$x^{5} dx = \frac{du}{6}$$$.

Also,

$${\color{red}{\int{x^{5} \left(x^{6} - 7\right)^{4} d x}}} = {\color{red}{\int{\frac{u^{4}}{6} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{6}$$$ und $$$f{\left(u \right)} = u^{4}$$$ an:

$${\color{red}{\int{\frac{u^{4}}{6} d u}}} = {\color{red}{\left(\frac{\int{u^{4} d u}}{6}\right)}}$$

Wenden Sie die Potenzregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=4$$$ an:

$$\frac{{\color{red}{\int{u^{4} d u}}}}{6}=\frac{{\color{red}{\frac{u^{1 + 4}}{1 + 4}}}}{6}=\frac{{\color{red}{\left(\frac{u^{5}}{5}\right)}}}{6}$$

Zur Erinnerung: $$$u=x^{6} - 7$$$:

$$\frac{{\color{red}{u}}^{5}}{30} = \frac{{\color{red}{\left(x^{6} - 7\right)}}^{5}}{30}$$

Daher,

$$\int{x^{5} \left(x^{6} - 7\right)^{4} d x} = \frac{\left(x^{6} - 7\right)^{5}}{30}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{x^{5} \left(x^{6} - 7\right)^{4} d x} = \frac{\left(x^{6} - 7\right)^{5}}{30}+C$$

$$$\int x^{5} \left(x^{6} - 7\right)^{4}\, dx = \frac{\left(x^{6} - 7\right)^{5}}{30} + C$$$A