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

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

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,

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

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

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

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

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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