Integral von $$$\frac{e^{\frac{x^{6}}{2}}}{x}$$$

Bestimme $$$\int \frac{e^{\frac{x^{6}}{2}}}{x}\, dx$$$.

Sei $$$u=x^{6}$$$.

Dann $$$du=\left(x^{6}\right)^{\prime }dx = 6 x^{5} dx$$$ (die Schritte sind » zu sehen), und es gilt $$$x^{5} dx = \frac{du}{6}$$$.

Also,

$${\color{red}{\int{\frac{e^{\frac{x^{6}}{2}}}{x} d x}}} = {\color{red}{\int{\frac{e^{\frac{u}{2}}}{6 u} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{6}$$$ und $$$f{\left(u \right)} = \frac{e^{\frac{u}{2}}}{u}$$$ an:

$${\color{red}{\int{\frac{e^{\frac{u}{2}}}{6 u} d u}}} = {\color{red}{\left(\frac{\int{\frac{e^{\frac{u}{2}}}{u} d u}}{6}\right)}}$$

Sei $$$v=\frac{u}{2}$$$.

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

Also,

$$\frac{{\color{red}{\int{\frac{e^{\frac{u}{2}}}{u} d u}}}}{6} = \frac{{\color{red}{\int{\frac{e^{v}}{v} d v}}}}{6}$$

Dieses Integral (Exponentialintegral) besitzt keine geschlossene Form:

$$\frac{{\color{red}{\int{\frac{e^{v}}{v} d v}}}}{6} = \frac{{\color{red}{\operatorname{Ei}{\left(v \right)}}}}{6}$$

Zur Erinnerung: $$$v=\frac{u}{2}$$$:

$$\frac{\operatorname{Ei}{\left({\color{red}{v}} \right)}}{6} = \frac{\operatorname{Ei}{\left({\color{red}{\left(\frac{u}{2}\right)}} \right)}}{6}$$

Zur Erinnerung: $$$u=x^{6}$$$:

$$\frac{\operatorname{Ei}{\left(\frac{{\color{red}{u}}}{2} \right)}}{6} = \frac{\operatorname{Ei}{\left(\frac{{\color{red}{x^{6}}}}{2} \right)}}{6}$$

Daher,

$$\int{\frac{e^{\frac{x^{6}}{2}}}{x} d x} = \frac{\operatorname{Ei}{\left(\frac{x^{6}}{2} \right)}}{6}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{e^{\frac{x^{6}}{2}}}{x} d x} = \frac{\operatorname{Ei}{\left(\frac{x^{6}}{2} \right)}}{6}+C$$

$$$\int \frac{e^{\frac{x^{6}}{2}}}{x}\, dx = \frac{\operatorname{Ei}{\left(\frac{x^{6}}{2} \right)}}{6} + C$$$A