Integral von $$$\frac{1}{x \ln\left(x^{3}\right)}$$$

Bestimme $$$\int \frac{1}{3 x \ln\left(x\right)}\, dx$$$.

Die Eingabe wird umgeschrieben: $$$\int{\frac{1}{x \ln{\left(x^{3} \right)}} d x}=\int{\frac{1}{3 x \ln{\left(x \right)}} d x}$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=\frac{1}{3}$$$ und $$$f{\left(x \right)} = \frac{1}{x \ln{\left(x \right)}}$$$ an:

$${\color{red}{\int{\frac{1}{3 x \ln{\left(x \right)}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{x \ln{\left(x \right)}} d x}}{3}\right)}}$$

Sei $$$u=\ln{\left(x \right)}$$$.

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

Somit,

$$\frac{{\color{red}{\int{\frac{1}{x \ln{\left(x \right)}} d x}}}}{3} = \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{3}$$

Das Integral von $$$\frac{1}{u}$$$ ist $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$\frac{{\color{red}{\int{\frac{1}{u} d u}}}}{3} = \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{3}$$

Zur Erinnerung: $$$u=\ln{\left(x \right)}$$$:

$$\frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{3} = \frac{\ln{\left(\left|{{\color{red}{\ln{\left(x \right)}}}}\right| \right)}}{3}$$

Daher,

$$\int{\frac{1}{3 x \ln{\left(x \right)}} d x} = \frac{\ln{\left(\left|{\ln{\left(x \right)}}\right| \right)}}{3}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{1}{3 x \ln{\left(x \right)}} d x} = \frac{\ln{\left(\left|{\ln{\left(x \right)}}\right| \right)}}{3}+C$$

$$$\int \frac{1}{3 x \ln\left(x\right)}\, dx = \frac{\ln\left(\left|{\ln\left(x\right)}\right|\right)}{3} + C$$$A