Integral von $$$x e^{x^{2}}$$$

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

Sei $$$u=x^{2}$$$.

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

Somit,

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

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(u \right)} = e^{u}$$$ an:

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

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

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

Zur Erinnerung: $$$u=x^{2}$$$:

$$\frac{e^{{\color{red}{u}}}}{2} = \frac{e^{{\color{red}{x^{2}}}}}{2}$$

Daher,

$$\int{x e^{x^{2}} d x} = \frac{e^{x^{2}}}{2}$$

Fügen Sie die Integrationskonstante hinzu:

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

$$$\int x e^{x^{2}}\, dx = \frac{e^{x^{2}}}{2} + C$$$A