Integral von $$$e^{\frac{\pi t}{2}}$$$

Bestimme $$$\int e^{\frac{\pi t}{2}}\, dt$$$.

Sei $$$u=\frac{\pi t}{2}$$$.

Dann $$$du=\left(\frac{\pi t}{2}\right)^{\prime }dt = \frac{\pi}{2} dt$$$ (die Schritte sind » zu sehen), und es gilt $$$dt = \frac{2 du}{\pi}$$$.

Daher,

$${\color{red}{\int{e^{\frac{\pi t}{2}} d t}}} = {\color{red}{\int{\frac{2 e^{u}}{\pi} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{2}{\pi}$$$ und $$$f{\left(u \right)} = e^{u}$$$ an:

$${\color{red}{\int{\frac{2 e^{u}}{\pi} d u}}} = {\color{red}{\left(\frac{2 \int{e^{u} d u}}{\pi}\right)}}$$

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

$$\frac{2 {\color{red}{\int{e^{u} d u}}}}{\pi} = \frac{2 {\color{red}{e^{u}}}}{\pi}$$

Zur Erinnerung: $$$u=\frac{\pi t}{2}$$$:

$$\frac{2 e^{{\color{red}{u}}}}{\pi} = \frac{2 e^{{\color{red}{\left(\frac{\pi t}{2}\right)}}}}{\pi}$$

Daher,

$$\int{e^{\frac{\pi t}{2}} d t} = \frac{2 e^{\frac{\pi t}{2}}}{\pi}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{e^{\frac{\pi t}{2}} d t} = \frac{2 e^{\frac{\pi t}{2}}}{\pi}+C$$

$$$\int e^{\frac{\pi t}{2}}\, dt = \frac{2 e^{\frac{\pi t}{2}}}{\pi} + C$$$A