Integral von $$$220 e^{\frac{x}{10}}$$$

Bestimme $$$\int 220 e^{\frac{x}{10}}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=220$$$ und $$$f{\left(x \right)} = e^{\frac{x}{10}}$$$ an:

$${\color{red}{\int{220 e^{\frac{x}{10}} d x}}} = {\color{red}{\left(220 \int{e^{\frac{x}{10}} d x}\right)}}$$

Sei $$$u=\frac{x}{10}$$$.

Dann $$$du=\left(\frac{x}{10}\right)^{\prime }dx = \frac{dx}{10}$$$ (die Schritte sind » zu sehen), und es gilt $$$dx = 10 du$$$.

Also,

$$220 {\color{red}{\int{e^{\frac{x}{10}} d x}}} = 220 {\color{red}{\int{10 e^{u} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=10$$$ und $$$f{\left(u \right)} = e^{u}$$$ an:

$$220 {\color{red}{\int{10 e^{u} d u}}} = 220 {\color{red}{\left(10 \int{e^{u} d u}\right)}}$$

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

$$2200 {\color{red}{\int{e^{u} d u}}} = 2200 {\color{red}{e^{u}}}$$

Zur Erinnerung: $$$u=\frac{x}{10}$$$:

$$2200 e^{{\color{red}{u}}} = 2200 e^{{\color{red}{\left(\frac{x}{10}\right)}}}$$

Daher,

$$\int{220 e^{\frac{x}{10}} d x} = 2200 e^{\frac{x}{10}}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{220 e^{\frac{x}{10}} d x} = 2200 e^{\frac{x}{10}}+C$$

$$$\int 220 e^{\frac{x}{10}}\, dx = 2200 e^{\frac{x}{10}} + C$$$A