Integral von $$$\frac{\pi x \sin{\left(x \right)}}{30}$$$

Bestimme $$$\int \frac{\pi x \sin{\left(x \right)}}{30}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=\frac{\pi}{30}$$$ und $$$f{\left(x \right)} = x \sin{\left(x \right)}$$$ an:

$${\color{red}{\int{\frac{\pi x \sin{\left(x \right)}}{30} d x}}} = {\color{red}{\left(\frac{\pi \int{x \sin{\left(x \right)} d x}}{30}\right)}}$$

Für das Integral $$$\int{x \sin{\left(x \right)} d x}$$$ verwenden Sie die partielle Integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Seien $$$\operatorname{u}=x$$$ und $$$\operatorname{dv}=\sin{\left(x \right)} dx$$$.

Dann gilt $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (Rechenschritte siehe ») und $$$\operatorname{v}=\int{\sin{\left(x \right)} d x}=- \cos{\left(x \right)}$$$ (Rechenschritte siehe »).

Das Integral wird zu

$$\frac{\pi {\color{red}{\int{x \sin{\left(x \right)} d x}}}}{30}=\frac{\pi {\color{red}{\left(x \cdot \left(- \cos{\left(x \right)}\right)-\int{\left(- \cos{\left(x \right)}\right) \cdot 1 d x}\right)}}}{30}=\frac{\pi {\color{red}{\left(- x \cos{\left(x \right)} - \int{\left(- \cos{\left(x \right)}\right)d x}\right)}}}{30}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=-1$$$ und $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ an:

$$\frac{\pi \left(- x \cos{\left(x \right)} - {\color{red}{\int{\left(- \cos{\left(x \right)}\right)d x}}}\right)}{30} = \frac{\pi \left(- x \cos{\left(x \right)} - {\color{red}{\left(- \int{\cos{\left(x \right)} d x}\right)}}\right)}{30}$$

Das Integral des Kosinus ist $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

$$\frac{\pi \left(- x \cos{\left(x \right)} + {\color{red}{\int{\cos{\left(x \right)} d x}}}\right)}{30} = \frac{\pi \left(- x \cos{\left(x \right)} + {\color{red}{\sin{\left(x \right)}}}\right)}{30}$$

Daher,

$$\int{\frac{\pi x \sin{\left(x \right)}}{30} d x} = \frac{\pi \left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{30}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{\pi x \sin{\left(x \right)}}{30} d x} = \frac{\pi \left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{30}+C$$

$$$\int \frac{\pi x \sin{\left(x \right)}}{30}\, dx = \frac{\pi \left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{30} + C$$$A