Integral von $$$3 x^{4} e^{x^{5}}$$$

Bestimme $$$\int 3 x^{4} e^{x^{5}}\, dx$$$.

Sei $$$u=x^{5}$$$.

Dann $$$du=\left(x^{5}\right)^{\prime }dx = 5 x^{4} dx$$$ (die Schritte sind » zu sehen), und es gilt $$$x^{4} dx = \frac{du}{5}$$$.

Somit,

$${\color{red}{\int{3 x^{4} e^{x^{5}} d x}}} = {\color{red}{\int{\frac{3 e^{u}}{5} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{3}{5}$$$ und $$$f{\left(u \right)} = e^{u}$$$ an:

$${\color{red}{\int{\frac{3 e^{u}}{5} d u}}} = {\color{red}{\left(\frac{3 \int{e^{u} d u}}{5}\right)}}$$

Das Integral der Exponentialfunktion lautet $$$\int{e^{u} d u} = e^{u}$$$:

$$\frac{3 {\color{red}{\int{e^{u} d u}}}}{5} = \frac{3 {\color{red}{e^{u}}}}{5}$$

Zur Erinnerung: $$$u=x^{5}$$$:

$$\frac{3 e^{{\color{red}{u}}}}{5} = \frac{3 e^{{\color{red}{x^{5}}}}}{5}$$

Daher,

$$\int{3 x^{4} e^{x^{5}} d x} = \frac{3 e^{x^{5}}}{5}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{3 x^{4} e^{x^{5}} d x} = \frac{3 e^{x^{5}}}{5}+C$$

$$$\int 3 x^{4} e^{x^{5}}\, dx = \frac{3 e^{x^{5}}}{5} + C$$$A