Integral von $$$e^{- \frac{5 x}{6}}$$$

Bestimme $$$\int e^{- \frac{5 x}{6}}\, dx$$$.

Sei $$$u=- \frac{5 x}{6}$$$.

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

Also,

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

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=- \frac{6}{5}$$$ und $$$f{\left(u \right)} = e^{u}$$$ an:

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

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

$$- \frac{6 {\color{red}{\int{e^{u} d u}}}}{5} = - \frac{6 {\color{red}{e^{u}}}}{5}$$

Zur Erinnerung: $$$u=- \frac{5 x}{6}$$$:

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

Daher,

$$\int{e^{- \frac{5 x}{6}} d x} = - \frac{6 e^{- \frac{5 x}{6}}}{5}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{e^{- \frac{5 x}{6}} d x} = - \frac{6 e^{- \frac{5 x}{6}}}{5}+C$$

$$$\int e^{- \frac{5 x}{6}}\, dx = - \frac{6 e^{- \frac{5 x}{6}}}{5} + C$$$A