Integral von $$$4 \pi \sin{\left(\pi x \right)}$$$

Bestimme $$$\int 4 \pi \sin{\left(\pi x \right)}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=4 \pi$$$ und $$$f{\left(x \right)} = \sin{\left(\pi x \right)}$$$ an:

$${\color{red}{\int{4 \pi \sin{\left(\pi x \right)} d x}}} = {\color{red}{\left(4 \pi \int{\sin{\left(\pi x \right)} d x}\right)}}$$

Sei $$$u=\pi x$$$.

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

Daher,

$$4 \pi {\color{red}{\int{\sin{\left(\pi x \right)} d x}}} = 4 \pi {\color{red}{\int{\frac{\sin{\left(u \right)}}{\pi} d u}}}$$

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

$$4 \pi {\color{red}{\int{\frac{\sin{\left(u \right)}}{\pi} d u}}} = 4 \pi {\color{red}{\frac{\int{\sin{\left(u \right)} d u}}{\pi}}}$$

Das Integral des Sinus lautet $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$4 {\color{red}{\int{\sin{\left(u \right)} d u}}} = 4 {\color{red}{\left(- \cos{\left(u \right)}\right)}}$$

Zur Erinnerung: $$$u=\pi x$$$:

$$- 4 \cos{\left({\color{red}{u}} \right)} = - 4 \cos{\left({\color{red}{\pi x}} \right)}$$

Daher,

$$\int{4 \pi \sin{\left(\pi x \right)} d x} = - 4 \cos{\left(\pi x \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{4 \pi \sin{\left(\pi x \right)} d x} = - 4 \cos{\left(\pi x \right)}+C$$

$$$\int 4 \pi \sin{\left(\pi x \right)}\, dx = - 4 \cos{\left(\pi x \right)} + C$$$A