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Integral von $$$x e^{2} \sin{\left(x \right)}$$$
Verwandter Rechner: Rechner für bestimmte und uneigentliche Integrale
Ihre Eingabe
Bestimme $$$\int x e^{2} \sin{\left(x \right)}\, dx$$$.
Lösung
Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=e^{2}$$$ und $$$f{\left(x \right)} = x \sin{\left(x \right)}$$$ an:
$${\color{red}{\int{x e^{2} \sin{\left(x \right)} d x}}} = {\color{red}{e^{2} \int{x \sin{\left(x \right)} d x}}}$$
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 »).
Also,
$$e^{2} {\color{red}{\int{x \sin{\left(x \right)} d x}}}=e^{2} {\color{red}{\left(x \cdot \left(- \cos{\left(x \right)}\right)-\int{\left(- \cos{\left(x \right)}\right) \cdot 1 d x}\right)}}=e^{2} {\color{red}{\left(- x \cos{\left(x \right)} - \int{\left(- \cos{\left(x \right)}\right)d x}\right)}}$$
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:
$$e^{2} \left(- x \cos{\left(x \right)} - {\color{red}{\int{\left(- \cos{\left(x \right)}\right)d x}}}\right) = e^{2} \left(- x \cos{\left(x \right)} - {\color{red}{\left(- \int{\cos{\left(x \right)} d x}\right)}}\right)$$
Das Integral des Kosinus ist $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:
$$e^{2} \left(- x \cos{\left(x \right)} + {\color{red}{\int{\cos{\left(x \right)} d x}}}\right) = e^{2} \left(- x \cos{\left(x \right)} + {\color{red}{\sin{\left(x \right)}}}\right)$$
Daher,
$$\int{x e^{2} \sin{\left(x \right)} d x} = \left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right) e^{2}$$
Fügen Sie die Integrationskonstante hinzu:
$$\int{x e^{2} \sin{\left(x \right)} d x} = \left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right) e^{2}+C$$
Antwort
$$$\int x e^{2} \sin{\left(x \right)}\, dx = \left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right) e^{2} + C$$$A