Integral von $$$\frac{k \sin{\left(\frac{x}{k} \right)}}{x}$$$ nach $$$x$$$

Bestimme $$$\int \frac{k \sin{\left(\frac{x}{k} \right)}}{x}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=k$$$ und $$$f{\left(x \right)} = \frac{\sin{\left(\frac{x}{k} \right)}}{x}$$$ an:

$${\color{red}{\int{\frac{k \sin{\left(\frac{x}{k} \right)}}{x} d x}}} = {\color{red}{k \int{\frac{\sin{\left(\frac{x}{k} \right)}}{x} d x}}}$$

Sei $$$u=\frac{x}{k}$$$.

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

Also,

$$k {\color{red}{\int{\frac{\sin{\left(\frac{x}{k} \right)}}{x} d x}}} = k {\color{red}{\int{\frac{\sin{\left(u \right)}}{u} d u}}}$$

Dieses Integral (Sinusintegral) besitzt keine geschlossene Form:

$$k {\color{red}{\int{\frac{\sin{\left(u \right)}}{u} d u}}} = k {\color{red}{\operatorname{Si}{\left(u \right)}}}$$

Zur Erinnerung: $$$u=\frac{x}{k}$$$:

$$k \operatorname{Si}{\left({\color{red}{u}} \right)} = k \operatorname{Si}{\left({\color{red}{\frac{x}{k}}} \right)}$$

Daher,

$$\int{\frac{k \sin{\left(\frac{x}{k} \right)}}{x} d x} = k \operatorname{Si}{\left(\frac{x}{k} \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{k \sin{\left(\frac{x}{k} \right)}}{x} d x} = k \operatorname{Si}{\left(\frac{x}{k} \right)}+C$$

$$$\int \frac{k \sin{\left(\frac{x}{k} \right)}}{x}\, dx = k \operatorname{Si}{\left(\frac{x}{k} \right)} + C$$$A