Integral von $$$e^{\frac{x}{5}}$$$

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

Sei $$$u=\frac{x}{5}$$$.

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

Also,

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

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

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

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

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

Zur Erinnerung: $$$u=\frac{x}{5}$$$:

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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