Integral von $$$e^{\frac{t}{50}}$$$

Bestimme $$$\int e^{\frac{t}{50}}\, dt$$$.

Sei $$$u=\frac{t}{50}$$$.

Dann $$$du=\left(\frac{t}{50}\right)^{\prime }dt = \frac{dt}{50}$$$ (die Schritte sind » zu sehen), und es gilt $$$dt = 50 du$$$.

Also,

$${\color{red}{\int{e^{\frac{t}{50}} d t}}} = {\color{red}{\int{50 e^{u} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=50$$$ und $$$f{\left(u \right)} = e^{u}$$$ an:

$${\color{red}{\int{50 e^{u} d u}}} = {\color{red}{\left(50 \int{e^{u} d u}\right)}}$$

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

$$50 {\color{red}{\int{e^{u} d u}}} = 50 {\color{red}{e^{u}}}$$

Zur Erinnerung: $$$u=\frac{t}{50}$$$:

$$50 e^{{\color{red}{u}}} = 50 e^{{\color{red}{\left(\frac{t}{50}\right)}}}$$

Daher,

$$\int{e^{\frac{t}{50}} d t} = 50 e^{\frac{t}{50}}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{e^{\frac{t}{50}} d t} = 50 e^{\frac{t}{50}}+C$$

$$$\int e^{\frac{t}{50}}\, dt = 50 e^{\frac{t}{50}} + C$$$A