Integral von $$$\frac{21 \sin{\left(\pi x \right)}}{2}$$$

Bestimme $$$\int \frac{21 \sin{\left(\pi x \right)}}{2}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=\frac{21}{2}$$$ und $$$f{\left(x \right)} = \sin{\left(\pi x \right)}$$$ an:

$${\color{red}{\int{\frac{21 \sin{\left(\pi x \right)}}{2} d x}}} = {\color{red}{\left(\frac{21 \int{\sin{\left(\pi x \right)} d x}}{2}\right)}}$$

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

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

Das Integral lässt sich umschreiben als

$$\frac{21 {\color{red}{\int{\sin{\left(\pi x \right)} d x}}}}{2} = \frac{21 {\color{red}{\int{\frac{\sin{\left(u \right)}}{\pi} d u}}}}{2}$$

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:

$$\frac{21 {\color{red}{\int{\frac{\sin{\left(u \right)}}{\pi} d u}}}}{2} = \frac{21 {\color{red}{\frac{\int{\sin{\left(u \right)} d u}}{\pi}}}}{2}$$

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

$$\frac{21 {\color{red}{\int{\sin{\left(u \right)} d u}}}}{2 \pi} = \frac{21 {\color{red}{\left(- \cos{\left(u \right)}\right)}}}{2 \pi}$$

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

$$- \frac{21 \cos{\left({\color{red}{u}} \right)}}{2 \pi} = - \frac{21 \cos{\left({\color{red}{\pi x}} \right)}}{2 \pi}$$

Daher,

$$\int{\frac{21 \sin{\left(\pi x \right)}}{2} d x} = - \frac{21 \cos{\left(\pi x \right)}}{2 \pi}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{21 \sin{\left(\pi x \right)}}{2} d x} = - \frac{21 \cos{\left(\pi x \right)}}{2 \pi}+C$$

$$$\int \frac{21 \sin{\left(\pi x \right)}}{2}\, dx = - \frac{21 \cos{\left(\pi x \right)}}{2 \pi} + C$$$A