Integral von $$$- \frac{\pi^{\pi} \sin{\left(x \right)}}{x}$$$

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

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

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

Dieses Integral (Sinusintegral) besitzt keine geschlossene Form:

$$- \pi^{\pi} {\color{red}{\int{\frac{\sin{\left(x \right)}}{x} d x}}} = - \pi^{\pi} {\color{red}{\operatorname{Si}{\left(x \right)}}}$$

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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