Integral von $$$e^{\frac{x^{2}}{8}}$$$

Bestimme $$$\int e^{\frac{x^{2}}{8}}\, dx$$$.

Sei $$$u=\frac{\sqrt{2} x}{4}$$$.

Dann $$$du=\left(\frac{\sqrt{2} x}{4}\right)^{\prime }dx = \frac{\sqrt{2}}{4} dx$$$ (die Schritte sind » zu sehen), und es gilt $$$dx = 2 \sqrt{2} du$$$.

Daher,

$${\color{red}{\int{e^{\frac{x^{2}}{8}} d x}}} = {\color{red}{\int{2 \sqrt{2} e^{u^{2}} d u}}}$$

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

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

Dieses Integral (Imaginäre Fehlerfunktion) besitzt keine geschlossene Form:

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

Zur Erinnerung: $$$u=\frac{\sqrt{2} x}{4}$$$:

$$\sqrt{2} \sqrt{\pi} \operatorname{erfi}{\left({\color{red}{u}} \right)} = \sqrt{2} \sqrt{\pi} \operatorname{erfi}{\left({\color{red}{\left(\frac{\sqrt{2} x}{4}\right)}} \right)}$$

Daher,

$$\int{e^{\frac{x^{2}}{8}} d x} = \sqrt{2} \sqrt{\pi} \operatorname{erfi}{\left(\frac{\sqrt{2} x}{4} \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{e^{\frac{x^{2}}{8}} d x} = \sqrt{2} \sqrt{\pi} \operatorname{erfi}{\left(\frac{\sqrt{2} x}{4} \right)}+C$$

$$$\int e^{\frac{x^{2}}{8}}\, dx = \sqrt{2} \sqrt{\pi} \operatorname{erfi}{\left(\frac{\sqrt{2} x}{4} \right)} + C$$$A