Integral von $$$e^{4 \theta}$$$

Bestimme $$$\int e^{4 \theta}\, d\theta$$$.

Sei $$$u=4 \theta$$$.

Dann $$$du=\left(4 \theta\right)^{\prime }d\theta = 4 d\theta$$$ (die Schritte sind » zu sehen), und es gilt $$$d\theta = \frac{du}{4}$$$.

Daher,

$${\color{red}{\int{e^{4 \theta} d \theta}}} = {\color{red}{\int{\frac{e^{u}}{4} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{4}$$$ und $$$f{\left(u \right)} = e^{u}$$$ an:

$${\color{red}{\int{\frac{e^{u}}{4} d u}}} = {\color{red}{\left(\frac{\int{e^{u} d u}}{4}\right)}}$$

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

$$\frac{{\color{red}{\int{e^{u} d u}}}}{4} = \frac{{\color{red}{e^{u}}}}{4}$$

Zur Erinnerung: $$$u=4 \theta$$$:

$$\frac{e^{{\color{red}{u}}}}{4} = \frac{e^{{\color{red}{\left(4 \theta\right)}}}}{4}$$

Daher,

$$\int{e^{4 \theta} d \theta} = \frac{e^{4 \theta}}{4}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{e^{4 \theta} d \theta} = \frac{e^{4 \theta}}{4}+C$$

$$$\int e^{4 \theta}\, d\theta = \frac{e^{4 \theta}}{4} + C$$$A