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

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

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

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

Für das Integral $$$\int{x e^{- \frac{6 x}{5}} d x}$$$ verwenden Sie die partielle Integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Seien $$$\operatorname{u}=x$$$ und $$$\operatorname{dv}=e^{- \frac{6 x}{5}} dx$$$.

Dann gilt $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (Rechenschritte siehe ») und $$$\operatorname{v}=\int{e^{- \frac{6 x}{5}} d x}=- \frac{5 e^{- \frac{6 x}{5}}}{6}$$$ (Rechenschritte siehe »).

Also,

$$5 {\color{red}{\int{x e^{- \frac{6 x}{5}} d x}}}=5 {\color{red}{\left(x \cdot \left(- \frac{5 e^{- \frac{6 x}{5}}}{6}\right)-\int{\left(- \frac{5 e^{- \frac{6 x}{5}}}{6}\right) \cdot 1 d x}\right)}}=5 {\color{red}{\left(- \frac{5 x e^{- \frac{6 x}{5}}}{6} - \int{\left(- \frac{5 e^{- \frac{6 x}{5}}}{6}\right)d x}\right)}}$$

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

$$- \frac{25 x e^{- \frac{6 x}{5}}}{6} - 5 {\color{red}{\int{\left(- \frac{5 e^{- \frac{6 x}{5}}}{6}\right)d x}}} = - \frac{25 x e^{- \frac{6 x}{5}}}{6} - 5 {\color{red}{\left(- \frac{5 \int{e^{- \frac{6 x}{5}} d x}}{6}\right)}}$$

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

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

Das Integral wird zu

$$- \frac{25 x e^{- \frac{6 x}{5}}}{6} + \frac{25 {\color{red}{\int{e^{- \frac{6 x}{5}} d x}}}}{6} = - \frac{25 x e^{- \frac{6 x}{5}}}{6} + \frac{25 {\color{red}{\int{\left(- \frac{5 e^{u}}{6}\right)d u}}}}{6}$$

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

$$- \frac{25 x e^{- \frac{6 x}{5}}}{6} + \frac{25 {\color{red}{\int{\left(- \frac{5 e^{u}}{6}\right)d u}}}}{6} = - \frac{25 x e^{- \frac{6 x}{5}}}{6} + \frac{25 {\color{red}{\left(- \frac{5 \int{e^{u} d u}}{6}\right)}}}{6}$$

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

$$- \frac{25 x e^{- \frac{6 x}{5}}}{6} - \frac{125 {\color{red}{\int{e^{u} d u}}}}{36} = - \frac{25 x e^{- \frac{6 x}{5}}}{6} - \frac{125 {\color{red}{e^{u}}}}{36}$$

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

$$- \frac{25 x e^{- \frac{6 x}{5}}}{6} - \frac{125 e^{{\color{red}{u}}}}{36} = - \frac{25 x e^{- \frac{6 x}{5}}}{6} - \frac{125 e^{{\color{red}{\left(- \frac{6 x}{5}\right)}}}}{36}$$

Daher,

$$\int{5 x e^{- \frac{6 x}{5}} d x} = - \frac{25 x e^{- \frac{6 x}{5}}}{6} - \frac{125 e^{- \frac{6 x}{5}}}{36}$$

Vereinfachen:

$$\int{5 x e^{- \frac{6 x}{5}} d x} = \frac{25 \left(- 6 x - 5\right) e^{- \frac{6 x}{5}}}{36}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{5 x e^{- \frac{6 x}{5}} d x} = \frac{25 \left(- 6 x - 5\right) e^{- \frac{6 x}{5}}}{36}+C$$

$$$\int 5 x e^{- \frac{6 x}{5}}\, dx = \frac{25 \left(- 6 x - 5\right) e^{- \frac{6 x}{5}}}{36} + C$$$A