Integral von $$$2 \sin{\left(\ln\left(x\right) \right)}$$$

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

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

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

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

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

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

Daher,

$$2 {\color{red}{\int{\sin{\left(\ln{\left(x \right)} \right)} d x}}}=2 {\color{red}{\left(\sin{\left(\ln{\left(x \right)} \right)} \cdot x-\int{x \cdot \frac{\cos{\left(\ln{\left(x \right)} \right)}}{x} d x}\right)}}=2 {\color{red}{\left(x \sin{\left(\ln{\left(x \right)} \right)} - \int{\cos{\left(\ln{\left(x \right)} \right)} d x}\right)}}$$

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

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

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

Das Integral lässt sich umschreiben als

$$2 x \sin{\left(\ln{\left(x \right)} \right)} - 2 {\color{red}{\int{\cos{\left(\ln{\left(x \right)} \right)} d x}}}=2 x \sin{\left(\ln{\left(x \right)} \right)} - 2 {\color{red}{\left(\cos{\left(\ln{\left(x \right)} \right)} \cdot x-\int{x \cdot \left(- \frac{\sin{\left(\ln{\left(x \right)} \right)}}{x}\right) d x}\right)}}=2 x \sin{\left(\ln{\left(x \right)} \right)} - 2 {\color{red}{\left(x \cos{\left(\ln{\left(x \right)} \right)} - \int{\left(- \sin{\left(\ln{\left(x \right)} \right)}\right)d x}\right)}}$$

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

$$2 x \sin{\left(\ln{\left(x \right)} \right)} - 2 x \cos{\left(\ln{\left(x \right)} \right)} + 2 {\color{red}{\int{\left(- \sin{\left(\ln{\left(x \right)} \right)}\right)d x}}} = 2 x \sin{\left(\ln{\left(x \right)} \right)} - 2 x \cos{\left(\ln{\left(x \right)} \right)} + 2 {\color{red}{\left(- \int{\sin{\left(\ln{\left(x \right)} \right)} d x}\right)}}$$

Wir sind bei einem Integral angelangt, das wir bereits gesehen haben.

Somit haben wir die folgende einfache Gleichung für das Integral erhalten:

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

Lösen wir es, erhalten wir, dass

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

Daher,

$$2 {\color{red}{\int{\sin{\left(\ln{\left(x \right)} \right)} d x}}} = 2 {\color{red}{\left(\frac{x \left(\sin{\left(\ln{\left(x \right)} \right)} - \cos{\left(\ln{\left(x \right)} \right)}\right)}{2}\right)}}$$

Daher,

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

Vereinfachen:

$$\int{2 \sin{\left(\ln{\left(x \right)} \right)} d x} = - \sqrt{2} x \cos{\left(\ln{\left(x \right)} + \frac{\pi}{4} \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{2 \sin{\left(\ln{\left(x \right)} \right)} d x} = - \sqrt{2} x \cos{\left(\ln{\left(x \right)} + \frac{\pi}{4} \right)}+C$$

$$$\int 2 \sin{\left(\ln\left(x\right) \right)}\, dx = - \sqrt{2} x \cos{\left(\ln\left(x\right) + \frac{\pi}{4} \right)} + C$$$A